Computer method and system for predicting physical properties using a conceptual segment model

ABSTRACT

Method of conducting chromatography comprising controlling a retention time of one or more chemical species in a mixture by determining at least one conceptual segment of: a) the one or more chemical species, b) a mobile phase component, and c) a stationary phase component. The method further includes defining an identity and an equivalent number of each of the at least one conceptual segment.

RELATED APPLICATIONS

This application is a divisional application of U.S. application Ser.No. 11/899,996, filed Sep. 7, 2007, which is a continuation-in-part ofU.S. application Ser. No. 11/528,749, filed Sep. 27, 2006, now U.S. Pat.No. 7,941,277, which is a continuation-in-part of U.S. application Ser.No. 11/241,675, filed Sep. 30, 2005, now U.S. Pat. No. 7,809,540, whichis a continuation-in-part of the U.S. application Ser. No. 10/785,925,filed Feb. 24, 2004, now U.S. Pat. No. 7,672,826. The entire teachingsof the above applications are incorporated herein by reference.

BACKGROUND OF THE INVENTION

Modeling physical properties of chemical mixtures is an important taskin many industries and processes. Specifically, for many processes,accurate modeling of physical properties for various mixtures is crucialfor such areas as process design and process control applications. Forexample, modeling physical properties of chemical mixtures is oftenuseful when selecting suitable solvents for use in chemical processes.

Solvent selection is an important task in the chemical synthesis andrecipe development phase of the pharmaceutical and agricultural chemicalindustries. The choice of solvent can have a direct impact on reactionrates, extraction efficiency, crystallization yield and productivity,etc. Improved solvent selection brings benefits, such as faster productseparation and purification, reduced solvent emission and lesser waste,lower overall costs, and improved production processes.

In choosing a solvent, various phase behavior characteristics of thesolvent-solute mixtures are considered. For example, vapor-liquidequilibrium (VLE) behavior is important when accounting for the emissionof solvent from reaction mixtures, and liquid-liquid miscibility (LLE)is important when a second solvent is used to extract target moleculesfrom the reaction media. For solubility calculations, solid-liquidequilibrium (SLE) is a key property when product isolation is donethrough crystallization at reduced temperature or with the addition ofanti-solvent.

For many applications, hundreds of typical solvents, not to mention analmost infinite number of mixtures thereof, are candidates in thesolvent selection process. In most cases, there is simply insufficientphase equilibrium data on which to make an informed solvent selection.For example, in pharmaceutical applications, it is often the case thatphase equilibrium data involving new drug molecules in the solventssimply do not exist. Although limited solubility experiments may betaken as part of the trial and error process, solvent selection islargely dictated by researchers' preferences or prior experiences.

Many solubility estimation techniques have been used to model thesolubility of components in chemical mixtures. Some examples include theHansen model and the UNIFAC group contribution model. Unfortunately,these models are rather inadequate because they have been developedmainly for petrochemicals with molecular weights in the 10s and the low100s daltons. These models do not extrapolate well for chemicals withlarger molecular weights, such as those encountered in pharmaceuticalapplications. Pharmaceuticals are mostly large, complex molecules withmolecular weight in the range of about 200-600 daltons.

Perhaps, the most commonly used methods in solvent selection process arethe solubility parameter models, i.e., the regular solution theory andthe Hansen solubility parameter model. There are no binary parameters inthese solubility parameter models and they all follow merely anempirical guide of “like dissolves like.” The regular solution model isapplicable to nonpolar solutions only, but not for solutions where polaror hydrogen-bonding interactions are significant. The Hansen modelextends the solubility parameter concept in terms of three partialsolubility parameters to better account for polar and hydrogen-bondingeffects.

In his book, Hansen published the solubility parameters for over 800solvents. See Hansen, C. M., HANSEN, SOLUBILITY PARAMETERS: A USER'SHANDBOOK (2000). Since Hansen's book contains the parameters for mostcommon solvents, the issue in using the Hansen model lies in thedetermination of the Hansen solubility parameters from regression ofavailable solubility data for the solute of interest in the solventselection process. Once determined, these Hansen parameters provide abasis for calculating activity coefficients and solubilities for thesolute in all the other solvents in the database. For pharmaceuticalprocess design, Bakken, et al. reported that the Hansen model can onlycorrelate solubility data with ±200% in accuracy, and it offers littlepredictive capability. See Bakken, et al., Solubility Modeling inPharmaceutical Process Design, paper presented at AspenTech User GroupMeeting, New Orleans, La., Oct. 5-8, 2003, and Paris, France, Oct.19-22, 2003.

When there are no data available, the UNIFAC functional groupcontribution method is sometimes used for solvent selection. Incomparison to the solubility parameter models, UNIFAC's strength comeswith its molecular thermodynamic foundation. It describes liquid phasenonideality of a mixture with the concept of functional groups. Allmolecules in the mixture are characterized with a set of pre-definedUNIFAC functional groups. The liquid phase nonideality is the result ofthe physical interactions between these functional groups and activitycoefficients of molecules are derived from those of functional groups,i.e., functional group additivity rule. These physical interactions havebeen pre-determined from available phase equilibrium data of systemscontaining these functional groups. UNIFAC gives adequate phaseequilibrium (VLE, LLE and SLE) predictions for mixtures with smallnonelectrolyte molecules as long as these molecules are composed of thepre-defined set of functional groups or similar groups.

UNIFAC fails for systems with large complex molecules for which eitherthe functional group additivity rule becomes invalid or due to undefinedUNIFAC functional groups. UNIFAC is also not applicable to ionicspecies, an important issue for pharmaceutical processes. Anotherdrawback with UNIFAC is that, even when valuable data become available,UNIFAC cannot be used to correlate the data. For pharmaceutical processdesign, Bakken et al., reported that the UNIFAC model only predictssolubilities with a RMS (root mean square) error on ln x of 2, or about±500% in accuracy, and it offers little practical value. Id.

A need exists for new, simple, and practical methods of accuratelymodeling one or more physical properties of a mixture of chemicals,including electrolytes.

SUMMARY OF THE INVENTION

The present invention provides an effective tool for the correlation andprediction of physical properties of a mixtures of chemical species,including electrolytes.

The present invention provides an effective tool for conductingchromatography based on the correlation and prediction of physicalproperties of a mixtures of chemical species.

In a first embodiment of the present invention, the present inventionfeatures methods of conducting chromatography. The methods comprisesteps of controlling a retention time of one or more chemical species ina mixture by determining at least one conceptual segment of: a) the oneor more chemical species, b) a mobile phase component and c) astationary phase component. The methods further include defining anidentity and an equivalent number of each of the at least one conceptualsegment.

In another embodiment, the methods further comprise a step of separatingthe one or more chemical species from the mixture.

In yet another embodiment, the methods include controlling a retentiontime of one or more chemical species in a mixture, and includescomputing at least one physical property for each of the one or morechemical species, the mobile phase component and the stationary phasecomponent. The at least one physical property is computed using thedetermined at least one conceptual segment and the equivalent number.

In a second embodiment, the present invention features methods ofconducting chromatography for separating one or more chemical speciesfrom a mixture. The methods comprise a step of assigning moleculardescriptors based on molecular interaction characteristics to each of:(i) the one or more chemical species, (ii) a mobile phase component, and(iii) a stationary phase component. The methods include measuringrespective molecular interaction indices for the one or more chemicalspecies, for the mobile phase component and for the stationary phasecomponent. The methods also include determining respective activitycoefficients of: (i) the one or more chemical species, (ii) the mobilephase component, and (iii) the stationary phase component, such that thedetermined respective activity coefficients enables conductingchromatography for separating the one or more chemical species from themixture.

In another embodiment, measuring respective molecular interactionindices for the one or more chemical species, for the mobile phasecomponent and for the stationary phase component includes determiningthe molecular interaction indices based on measures of effective surfaceareas expressing surface molecular interaction characteristics for eachof the one or more chemical species, of the mobile phase component, andof the stationary phase component.

In another embodiment, assigning one or more predetermined moleculardescriptors based on molecular interactive characteristics includesassigning one or more predetermined conceptual segments to each of theone or more chemical species, of the mobile phase component and of thestationary phase component.

In another embodiment, measuring respective molecular interactionindices for the one or more chemical species, and the mobile phasecomponent and the stationary phase component includes determining anequivalent number for the predetermined conceptual segment.

In a third embodiment, the present invention is a computer programproduct. The computer program includes a computer usable medium, and aset of computer program instructions embodied on the computer usablemedium for conducting chromatography for separating a one or morechemical species from a mixture by:

a) assigning molecular descriptors based on molecular interactioncharacteristics to each of: (i) the one or more chemical species, (ii) amobile phase component, and (iii) a stationary phase component;

b) measuring respective molecular interaction indices for the one ormore chemical species, for the mobile phase component and for thestationary phase component; and c) determining respective activitycoefficients of: (i) the one or more chemical species; (ii) the mobilephase component; and (iii) the stationary phase component, such that thedetermined respective activity coefficients enables conductingchromatography for separating the one or more chemical species from themixture.

In a fourth embodiment, the present invention features a computer systemfor conducting chromatography. The computer system includes a user inputmeans for obtaining empirical data from a user, and a digital processorcoupled to receive the empirical data from the input means. The digitalprocessor executes a modeling system in working memory, and the modelingsystem uses the empirical data for assigning molecular descriptors basedon molecular interaction characteristics to each of: (i) each of the oneor more chemical species, (ii) a mobile phase component, and (iii) astationary phase component; measuring respective molecular interactiveindices for the one or more chemical species, and the mobile phasecomponent and for the stationary phase component, and determiningactivity coefficients of: (i) the one or more chemical species, (ii) themobile phase component, and (iii) the stationary phase component suchthat the determined respective activity coefficients enables conductingchromatography for separating the one or more chemical species from themixture.

In one embodiment of the computer system, the chromatography includesreverse-phase chromatography, normal phase chromatography, hydrophobicinteraction chromatography, hydrophilic interaction chromatography,ion-pair chromatography, or ion-exchange chromatography or otherchromatography.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features and advantages of theinvention will be apparent from the following more particulardescription of preferred embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingthe principles of the invention.

FIG. 1 is a schematic view of a computer network in which the presentinvention may be implemented.

FIG. 2 is a block diagram of a computer of the network of FIG. 1.

FIGS. 3-4 b are flow diagrams of one embodiment of the present inventionemployed in the computer network environment of FIGS. 1 and 2.

FIG. 5 illustrates a graph showing the binary phase diagram for a water,1,4-dioxane mixture at atmospheric pressure.

FIG. 6 illustrates a graph showing the binary phase diagram for a water,octanol mixture at atmospheric pressure.

FIG. 7 illustrates a graph showing the binary phase diagram for anoctanol, 1,4-dioxane mixture at atmospheric pressure.

FIG. 8 illustrates a graph showing data of experimental solubilities vs.calculated solubilities for p-aminobenzoic acid in various solvents at298.15K.

FIG. 9 illustrates a graph showing data of experimental solubilities vs.calculated solubilities for benzoic acid in various solvents at 298.15K.

FIG. 10 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for camphor in various solvents at 298.15K.

FIG. 11 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for ephedrine in various solvents at298.15K.

FIG. 12 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for lidocaine in various solvents at298.15K.

FIG. 13 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for methylparaben in various solvents at298.15K.

FIG. 14 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for testosterone in various solvents at298.15K.

FIG. 15 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for theophylline in various solvents at298.15K.

FIG. 16 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for estriol in nine solvents at 298.15K.

FIG. 17 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for estrone in various solvents at 298.15K.

FIG. 18 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for morphine in six solvents at 308.15K.

FIG. 19 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for piroxicam in 14 solvents at 298.15K.

FIG. 20 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for hydrocortisone in 11 solvents at298.15K.

FIG. 21 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for haloperidol in 13 solvents at 298.15K.

FIG. 22 is a graph illustrating the effect of hydrophobicity parameter Xon natural logarithm of mean ionic activity coefficient of aqueouselectrolytes with E=1.

FIG. 23 is a graph illustrating the effect of polarity parameter Y− onnatural logarithm of mean ionic activity coefficient of aqueouselectrolytes with E=1.

FIG. 24 is a graph illustrating the effect of polarity parameter Y+ onnatural logarithm of mean ionic activity coefficient of aqueouselectrolytes with E=1.

FIG. 25 is a graph illustrating the effect of hydrophilicity parameter Zon natural logarithm of mean ionic activity coefficient of aqueouselectrolytes with E=1.

FIG. 26 is a graph illustrating the effect of electrolyte parameter E onnatural logarithm of mean ionic activity coefficient of aqueouselectrolytes.

FIG. 27 is a graph illustrating comparison of experimental andcalculated molality scale mean ionic activity coefficients ofrepresentative aqueous electrolytes at 298.15 K.

FIG. 28 is a graph illustrating the present invention model results forsodium chloride solubility at 298.15 K.

FIG. 29 is a graph illustrating the present invention model results forsodium acetate solubility at 298.15 K.

FIG. 30 a is a graph illustrating the present invention model resultsfor benzoic acid solubility at 298.15 K.

FIG. 30 b is a graph illustrating the present invention model resultsfor sodium benzoate solubility at 298.15 K.

FIG. 31 a is a graph illustrating the present invention model resultsfor salicylic acid solubility at 298.15 K.

FIG. 31 b is a graph illustrating the present invention model resultsfor sodium salicylate solubility at 298.15 K.

FIG. 32 a is a graph illustrating the present invention model resultsfor p-aminobenzoic acid solubility at 298.15 K.

FIG. 32 b is a graph illustrating the present invention model resultsfor sodium p-aminobenzoate solubility at 298.15 K.

FIG. 33 a is a graph illustrating the present invention model resultsfor ibuprofen solubility at 298.15 K

FIG. 33 b is a graph illustrating the present invention model resultsfor sodium ibuprofen solubility at 298.15 K.

FIG. 34 a is a graph illustrating the present invention model resultsfor diclofenac solubility at 298.15 K.

FIG. 34 b is a graph illustrating the present invention model resultsfor sodium diclofenac solubility at 298.15 K.

DETAILED DESCRIPTION OF THE INVENTION

A description of example embodiments of the invention follows.

The terms “conceptual segments” refer to molecular descriptors, based onexpressed characteristics of molecular interactions between two species(e.g., a solute species in a solution interacting with the surface ofthe stationary phase in a chromatography processes) as one skilled inthe art would understand. In certain embodiments, conceptual segmentsdescribe for each solute and solvent molecule their effective surfaceinteraction in terms of the following types of the conceptual segments:hydrophobic segment, polar segment, hydrophilic segment, and solvationsegment. The conceptual segments are to simulate the interactioncharacteristics of representative molecular surfaces. For example, thehydrophilic segment simulates polar molecular surfaces that are“hydrogen bond donor or acceptor.” As such, it represents molecularsurfaces with the tendency to form a hydrogen bond. Conversely, thehydrophobic segment simulates molecular surfaces with the adversity toform a hydrogen bond. As for the polar segment, it simulates polarmolecular surfaces that are “electron pair donor or acceptor.” With theconceptual segments identified, real molecules are then selected asreference molecules for the conceptual segments and available phaseequilibrium data of these reference molecules are used to identifymolecular interaction energy parameters for the conceptual segments. Thereference molecules with distinct molecular characteristics (i.e.,hydrophobic, hydrophilic, or polar) and with abundant, publiclyavailable phase equilibrium data are chosen for practical purposes.

The term “equivalent numbers” of the conceptual segments, which isinterchangeably used with “conceptual segment numbers” in thisapplication, refers to measures of the surface areas of a molecule thatare associated with a molecular interaction with another molecule. Themolecular interaction can be characterized according to the types of theconceptual segments such as hydrophobicity, polarity, hydrophilicity andsolvation. These measures of the molecular interaction can be determinedfrom methods known by one skilled in the art including the interactioncharacteristics of the molecules in solution as expressed in availableexperimental data such as their experimental phase equilibrium data.

The present invention provides a new system and method for modeling thephysical properties or behavior of chemical mixtures (e.g., chemicalsolutions or suspensions). Briefly, the molecular structure of one ormore species in a chemical mixture is assigned one or more differenttypes of “conceptual segments.” An equivalent number is determined foreach conceptual segment. This conceptual segment approach of the presentinvention is referred to as the Non-Random Two-Liquid Segment ActivityCoefficient (“NRTL-SAC”) model for nonelectrolytes and as theelectrolyte extension of NRTL-SAC (“eNRTL-SAC”) model for electrolytes.In some embodiments, this invention features methods of conductingindustrial manufacture, research or development. In one embodiment, themethods comprise computer implemented steps of modeling at least onephysical property of a mixture of at least two chemical species bydetermining at least one conceptual segment for each of the chemicalspecies. Determining at least one conceptual segment includes definingan identity and an equivalent number of each conceptual segment.

In one embodiment, the methods of conducting industrial manufacture,research or development further include the steps of: using thedetermined conceptual segments, computing at least one physical propertyof the mixture; and b) providing an analysis of the computed physicalproperty. The analysis forms a model of the at least one physicalproperty of the mixture.

In further embodiment, the method of the first embodiment that includesthe mixture includes more than one phase and at least a portion of atleast one chemical species is in a liquid phase. In one embodiment, themixture includes any number and combination of vapor, solid, and liquidphase. In some embodiment, the mixture includes at least one liquidphase and at least one solid phase. In yet another embodiment, themixture includes a first liquid phase, a second liquid phase, and afirst chemical species. At least a portion of the first chemical speciesis dissolved in both the first liquid phase and the second liquid phase.

In further embodiments, the methods of the first embodiment can computesolubility of at least one of the chemical species in at least one phaseof the mixture.

In further embodiments, the methods of the first embodiment can definethe identity that includes identifying each conceptual segment as one ofa hydrophobic segment, a hydrophilic segment, or a polar segment.

The methods of this invention can model a wide range of chemicalmixtures of nonelectrolytes and electrolytes. For example, the chemicalmixtures can include one or more of the following types of chemicalspecies: an electrolyte, an organic nonelectrolyte, an organic salt, acompound possessing a net charge, a zwitterions, a polar compound, anonpolar compound, a hydrophilic compound, a hydrophobic compound, apetrochemical, a hydrocarbon, a halogenated hydrocarbon, an ether, aketone, an ester, an amide, an alcohol, a glycol, an amine, an acid,water, an alkane, a surfactant, a polymer, and an oligomer.

In further embodiments, the mixture includes at least one chemicalspecies which is a solvent (e.g., a solvent used in a pharmaceuticalproduction, screening, or testing process), a solute, a pharmaceuticalcomponent, a compound used in an agricultural application (e.g., aherbicide, a pesticide, or a fertilizer) or a precursor of a compoundused in an agricultural application, a compound used in an adhesivecomposition or a precursor of a compound used in an adhesivecomposition, a compound used in an ink composition or a precursor of acompound used in an ink composition. As used herein, a “pharmaceuticalcomponent” includes a pharmaceutical compound, drug, therapeutic agent,or a precursor thereof (i.e., a compound used as an ingredient in apharmaceutical compound production process). The “pharmaceuticalcomponent” of this invention can be produced by any publicly knownmethod or by any method equivalent with the former. The pharmaceuticalagent or other active compound of the present invention may comprise asingle pharmaceutical ingredient or a combination of pharmaceuticalingredients, including active ingredients. These active ingredients maybe incorporated in the adhesive layer, backing layer or in both. Apharmaceutical component can also include ingredients for enhancing drugsolubility and/or stability of the drug to be added to the layer orlayers containing the active ingredient. In some embodiments, themixture includes at least one pharmaceutical component having amolecular weight greater than about 900 daltons, at least onepharmaceutical component having a molecular weight in the range ofbetween about 100 daltons and about 900 daltons, and/or at least onepharmaceutical component having a molecular weight in the range ofbetween about 200 daltons and about 600 daltons. In further embodiments,the mixture includes at least one nonpolymeric pharmaceutical component.

In further embodiments, the mixture includes at least one ICH solvent,which is a solvent listed in the ICH Harmonized Tripartite Guideline,Impurities: Guideline for Residual Solvents Q3C, incorporated herein inits entirety by reference. ICH STEERING COMMITTEE , ICH HarmonizedTripartite Guideline, Impurities: Guideline for Residual Solvents Q3C,International Conference of Harmonization of Technical Requirements forRegistration of Pharmaceuticals for Human Use (1997).

It will be apparent to those skilled in the art that a component of themixture can belong to more than one type of chemical species.

In accordance with one aspect of the present invention, at least oneconceptual segment (e.g., at least 1, 2, 3, 4, 5, 7, 10, 12, or morethan 12 conceptual segments) is determined or defined for each of thechemical species of the mixture. The conceptual segments are moleculardescriptors of the various molecular species in the mixture. An identityand an equivalent number are determined for each of the conceptualsegments. Examples of identities for conceptual segments include ahydrophobic segment, a polar segment, a hydrophilic segment, a chargedsegment, and the like. Experimental phase equilibrium data can be usedto determine the equivalent number of the conceptual segment(s).

The determined conceptual segments are used to compute at least onephysical property of the mixture, and an analysis of the computedphysical property is provided to form a model of at least one physicalproperty of the mixture. The methods of this invention are able to modela wide variety of physical properties. Examples of physical propertiesinclude vapor pressure, solubility (e.g., the equilibrium concentrationof one or more chemical species in one or more phases of the mixture),boiling point, freezing point, octanol/water partition coefficient,lipophilicity, and other physical properties that are measured ordetermined for use in the chemical processes.

Preferably, the methods provide equilibrium values of the physicalproperties modeled. For example, a mixture can include at least oneliquid solvent and at least one solid pharmaceutical component and themethods can be used to model the solubility of the pharmaceuticalcomponent. In this way, the methods can provide the concentration of theamount (e.g., a concentration value) of the pharmaceutical componentthat will be dissolved in the solvent at equilibrium. In anotherexample, the methods can model a mixture that includes a solid phase(e.g., a solid pharmaceutical component) and at least two liquid phases(e.g., two solvent that are immiscible in one another). The model canpredict, or be used to predict, how much of the pharmaceutical componentwill be dissolved in the two liquid phases and how much will be left inthe solid phase at equilibrium. In yet a further embodiment, the methodscan be used to predict the behavior of a mixture after a change hasoccurred. For example, if the mixture includes two liquid phases and onesolid phase, and an additional chemical species is introduced into themixture (e.g., a solvent, pharmaceutical component, or other chemicalcompound), additional amounts of a chemical species are introduced intothe mixture, and/or one or more environmental conditions are changes(e.g., a change in temperature and/or pressure), the method can be usedto predict how the introduction of the chemical species and/or change inconditions will alter one or more physical properties of the mixture atequilibrium.

The models of the physical property or properties of the mixture areproduced by determining the interaction characteristics of theconceptual segments. In some embodiments, the segment-segmentinteraction characteristics of the conceptual segments are representedby their corresponding binary NRTL parameters. (See Example 11.) Giventhe NRTL parameters for the conceptual segments and the numbers andtypes of conceptual segments for the molecules, the NRTL-SAC modelcomputes activity coefficients for the segments and then for the variousmolecules in the mixture. In other words, the physical properties orbehavior of the mixture will be accounted for based on the segmentcompositions of the molecules and their mutual interactions. Theactivity coefficient of each molecule is computed from the number andtype of segments for each molecule and the corresponding segmentactivity coefficients.

In one embodiment, the invention features methods of conductingindustrial manufacture, research or development where the at least twochemical species includes at least one electrolyte. Electrolytesdissociate to ionic species in solutions. For “strong” electrolytes, thedissociation is “completely” to ionic species. For “weak” electrolytes,the dissociation is partially to ionic species while undissociatedelectrolytes, similar to nonelectrolytes, remain as neutral molecularspecies. Complexation of ionic species with solvent molecules or otherionic species may also occur. An implication of the electrolyte solutionchemistry is that the extended model should provide a thermodynamicallyconsistent framework to compute activity coefficients for both molecularspecies and ionic species.

Preferably, the method comprises computer implemented steps of: (a)using the determined conceptual electrolyte segment, computing at leastone physical property of the mixture; and (b) providing an analysis ofthe computed physical property. The analysis forms a model of the atleast one physical property of the mixture. The methods of thisinvention are able to model a wide variety of physical propertiesinvolving electrolytes, including activity coefficient, vapor pressure,solubility, boiling point, freezing point, octanol/water partitioncoefficient, and lipophilicity of the electrolyte.

The computed physical property of the analysis can include at least oneof activity coefficient, vapor pressure, solubility, boiling point,freezing point, octanol/water partition coefficient, and lipophilicityof the electrolyte.

In a more preferred embodiment, the step of computing at least onephysical property includes calculating the activity coefficient of theionic species derived from the electrolyte.

In further embodiments, the methods include the electrolyte that is anyone of a pharmaceutical compound, a nonpolymeric compound, a polymer, anoligomer, an inorganic compound and an organic compound. In someembodiment, the electrolyte is symmetrical or unsymmetrical. In anotherembodiment, the electrolyte is univalent or multivalent. In yet anotherembodiment, the electrolyte includes two or more ionic species.

In some embodiments, the invention features methods of conducting apharmaceutical activity. In one embodiment, the methods comprise thecomputer implemented steps of modeling at least one physical property ofa mixture of at least two chemical species by determining at least oneconceptual segment for each of the chemical species. Determining atleast one conceptual segment includes defining an identity and anequivalent number of each conceptual segment.

The term “pharmaceutical activity”, as used herein, has the meaningcommonly afforded the term in the art. A pharmaceutical activity caninclude ones for drug discovery, development or manufacture.Particularly, a pharmaceutical activity can include one that is art,practice, or profession of researching, preparing, preserving,compounding, and dispensing medical drugs and that is of, relating to,or engaged in pharmacy or the manufacture and sale of pharmaceuticals. Apharmaceutical activity further includes the branch of health/medicalscience and the sector of public life concerned with maintaining orrestoring human/mammalian health through the study, diagnosis andtreatment of disease and injury. It includes both an area ofknowledge—i.e. the chemical make-up of a drug—and the appliedpractice—i.e. drugs in relation to some diseases and methods oftreatment. A pharmaceutical activity can also include at least one ofdrug design, drug synthesis, drug formulation, drug characterization,drug screen and assay, clinical evaluation, and drug purification. In amore preferred embodiment, the drug synthesis can include distillation,screening, crystallization, filtration, washing, or drying.

In particular, the terms “drug design”, as used herein, has the meaningcommonly afforded the term in the art. Drug design can include theapproach of finding drugs by design, based on their biological targets.Typically, a drug target is a key molecule involved in a particularmetabolic or signaling pathway that is specific to a disease conditionor pathology, or to the infectivity or survival of a microbial pathogen.The term “drug characterization”, as used herein, also has the meaningcommonly afforded the term in the art. The meaning can include a widerange of analyses to obtain identity, purity, and stability data for newdrug substances and formulations, including: structural identity andconfirmation, certificates of analyses, purity determinations,stability-indicating methods development and validation identificationand quantification of impurities, and residual solvent analyses.

In some embodiments, the pharmaceutical activity can include studies ona molecular interaction within the mixture. The term “study”, usedherein, can include an endeavor for acquiring knowledge about a givensubject through, for example, an experiment, (i.e. clinical trial). In apreferred embodiment, examples of the studies can include one or more ofpharmacokinetics, pharmacodynamics, solvent screening, combination drugtherapy, drug toxicity, a process design for an active pharmaceuticalingredient, and chromatography. The cited types of study have themeaning commonly afforded the term in the art.

In further embodiments, the methods of conducting a pharmaceuticalactivity can comprise the mixture that includes at least one liquidphase. In one embodiment, the methods can include any number andcombination of vapor, solid and liquid phases. In another embodiment,the methods include at least one liquid phase and at least one solidphase. In a preferred embodiment, the mixture can include at least oneliquid solvent and at least one pharmaceutical component. In a morepreferred embodiment, the mixture can include more than one phase and atleast a portion of the at least one pharmaceutical component. Thepharmaceutical component can be an active pharmaceutical ingredient.

The liquid phase can be an amorphous phase. The term “an amorphousphase” used herein, has the meaning commonly afforded the term in theart. An amorphous phase can include a solid in which there is nolong-range order of the positions of the atoms. (Solids in which thereis long-range atomic order are called crystalline solids.) Most classesof solid materials can be found or prepared in an amorphous form. Forinstance, common window glass is an amorphous ceramic, many polymers(such as polystyrene) are amorphous, and even foods such as cotton candyare amorphous phase. Amorphous materials are often prepared by rapidlycooling molten material. The cooling reduces the mobility of thematerial's molecules before they can pack into a more thermodynamicallyfavorable crystalline state. Amorphous materials can also be produced byadditives which interfere with the ability of the primary constituent tocrystallize. For example addition of soda to silicon dioxide results inwindow glass and the addition of glycols to water results in a vitrifiedsolid. In a preferred embodiment, at least one of the species in themixture that is in the amorphous phase is an active pharmaceuticalingredient. In a more preferred embodiment, the method can include astep of estimating an amorphous phase solubility by calculating a phaseequilibrium between a solute rich phase and a solvent rich phase.

In some embodiments, the methods of conducting a pharmaceutical activitycan include a mixture that has at least one of the at least two chemicalspecies is a pharmaceutical component. In a preferred embodiment, thepharmaceutical component is an active pharmaceutical ingredient.

In some embodiments, the methods of conducting a pharmaceutical activitycan further comprise the steps of: (a) using the determined conceptualsegments, computing at least one physical property of the mixture; and(b) providing an analysis of the computed physical property. Theanalysis forms a model of the at least one physical property of themixture. In a preferred embodiment, the step of defining an identity caninclude steps of identifying each conceptual segment as one of ahydrophobic segment, a hydrophilic segment, a polar segment, or asolvation segment.

In some embodiments, the methods of conducting a pharmaceutical activitycan comprise a mixture of at least two chemical species that includes atleast one electrolyte. In further embodiments, the methods furtherinclude the steps of: a) using the determined conceptual electrolytesegment, computing at least one physical property of the mixture; and b)providing an analysis of the computed physical property. The analysisforms a model of the at least one physical property of the mixture. Inone embodiment, the step of computing at least one physical property caninclude steps of calculating the activity coefficient of the ionicspecies derived from the electrolyte. In a preferred embodiment, thecomputed physical property of the analysis can include at least one ofactivity coefficient, vapor pressure, solubility, boiling point,freezing point, octanol/water partition coefficient, and lipophilicityof the electrolyte.

In further embodiment, the conceptual electrolyte segment can include acationic segment and an anionic segment, both segments of unity ofcharge.

In some embodiment, the electrolyte is any one of: a pharmaceuticalcompound, a nonpolymeric compound, a polymer, an oligomer, an inorganiccompound and an organic compound. In one embodiment, the electrolyte issymmetrical or unsymmetrical. In another embodiment, the electrolyte isunivalent or multivalent. In yet another embodiment, the electrolyteincludes two or more ionic species.

In some embodiments, the present invention features methods ofseparating one or more chemical species from a mixture. The methodsinclude steps of modeling molecular interaction between the chemicalspecies in one or more solvents by determining at least one conceptualsegment for each of the species, including defining an identity and anequivalent number of each conceptual segment.

In further embodiments, the methods of separating one or more chemicalspecies from a mixture can use chromatography. In a preferredembodiment, the types of chromatography can include one of thefollowing: capillary-action chromatography, paper chromatography, thinlayer chromatography, column chromatography, fast protein liquidchromatography, high performance liquid chromatography, ion exchangechromatography, affinity chromatography, gas chromatography, andcountercurrent chromatography. In a more preferred embodiment, thechromatography is high performance liquid chromatography.

In one embodiment, the methods of separating one or more chemicalspecies can comprise a mixture that includes at least one liquid phase.In another embodiment, the method of separating one or more chemicalspecies can comprise a mixture that includes at least one liquid phaseand that at least a portion of at least one chemical species is in theliquid phase. In yet anther embodiment, at least one of the chemicalspecies of the method of separating one or more chemical species is anactive pharmaceutical ingredient.

In some embodiments, the methods of separating one or more chemicalspecies can include the steps of: a) using the determined conceptualsegments, computing at least one physical property of the mixture; andb) providing an analysis of the computed physical property. The analysisforms a model of the at least one physical property of the mixture. Infurther embodiments, the steps of defining an identity can include stepsof identifying each conceptual segment as one of a hydrophobic segment,a hydrophilic segment, a polar segment, or a solvation segment.

In some embodiments, this invention features computer program products.The computer program products comprise a computer usable medium and aset of computer program instructions embodied on the computer useablemedium for conducting industrial manufacture, research or development bymodeling at least one physical property of a mixture of at least twochemical species by determining at least one conceptual segment for eachof the chemical species. Included are instructions to define an identityand an equivalent number of each of conceptual segment. In a preferredembodiment, the computer usable medium can include a removable storagemedium. In a more preferred embodiment, the removable storage medium caninclude any of a CD-ROM, a DVD-ROM, a diskette, and a tape.

In further embodiment, the computer program products can include: (a)instructions to use the determined conceptual segments to compute atleast one physical property of the chemical mixture; and (b)instructions to provide an analysis of the computed physical property.The analysis forms a model of at least one physical property of themixture.

In one embodiment of the computer program products, at least someportion of the computer program instructions can include instructions torequest data or request instructions over a telecommunications network.In another embodiment, at least some portion of the computer program istransmitted over a global network.

In another embodiment of the computer program products, an industrialmanufacture, research or development can include a pharmaceuticalactivity. In a preferred embodiment, the pharmaceutical activity caninclude one or more of the following: pharmacokinetics,pharmacodynamics, solvent screening, crystallization productivity, drugformulation, combination drug therapy, drug toxicity, a process designfor an active pharmaceutical ingredient, capillary-actionchromatography, paper chromatography, thin layer chromatography, columnchromatography, fast protein liquid chromatography, high performanceliquid chromatography, ion exchange chromatography, affinitychromatography, gas chromatography, and countercurrent chromatography.

In some embodiment, the invention features computer systems forconducting industrial manufacture, research or development by modelingat least one physical property of a mixture of at least two chemicalspecies. The computer systems can include: a) a user input means forobtaining chemical data from a user; b) a digital processor coupled toreceive obtained chemical data input from the input means; and c) anoutput means coupled to the digital processor. The digital processorexecutes a modeling system in working memory, and the output meansprovides to the user the formed model of the physical property of themixture. The modeling system may use the chemical data to determine atleast one conceptual segment for each of the chemical species, includingdefining an identity and equivalent number of each conceptual segment.

In further embodiments, the computer system can: a) use the determinedconceptual electrolyte segment to compute at least one physical propertyof the mixture; and b) provide an analysis of the computed physicalproperty. The analysis forms a model of at least one physical propertyof the mixture. In a preferred embodiment, the computer system canenable transmission of some portion of at least one of the chemical dataand the formed model over a global network. Alternatively, the computersystem can also conduct industrial manufacture, research or developmentthat includes a pharmaceutical activity. In a preferred embodiment,conducting industrial manufacture, research or development can includeone or more of the following: pharmacokinetics, pharmacodynamics,solvent screening, crystallization productivity, drug formulation,combination drug therapy, drug toxicity, a process design for an activepharmaceutical ingredient. capillary-action chromatography, paperchromatography, thin layer chromatography, column chromatography, fastprotein liquid chromatography, high performance liquid chromatography,ion exchange chromatography, affinity chromatography, gaschromatography, and countercurrent chromatography.

Reference is now made to a preferred embodiment of the present inventionas illustrated in FIGS. 1-4. FIG. 1 illustrates a computer network orsimilar digital processing environment in which the present inventionmay be implemented.

Referring to FIG. 1, client computer(s)/devices 50 and servercomputer(s) 60 provide processing, storage, and input/output devicesexecuting application programs and the like. Client computer(s)/devices50 can also be linked through communications network 70 to othercomputing devices, including other client devices/processes 50 andserver computer(s) 60. Communications network 70 can be part of a remoteaccess network, a global network (e.g., the Internet), a worldwidecollection of computers, Local area or Wide area networks, and gatewaysthat currently use respective protocols (TCP/IP, Bluetooth, etc.) tocommunicate with one another. Other electronic device/computer networkarchitectures are suitable.

FIG. 2 is a diagram of the internal structure of a computer (e.g.,client processor/device 50 or server computers 60) in the computersystem of FIG. 1. Each computer 50, 60 contains system bus 79, where abus is a set of hardware lines used for data transfer among thecomponents of a computer or processing system. Bus 79 is a sharedconduit that connects different elements of a computer system (e.g.,processor, disk storage, memory, input/output ports, network ports,etc.) that enables the transfer of information between the elements.Attached to system bus 79 is I/O device interface 82 for connectingvarious input and output devices (e.g., keyboard, mouse, displays,printers, speakers, etc.) to the computer 50, 60. Network interface 86allows the computer to connect to various other devices attached to anetwork (e.g., network 70 of FIG. 1). Memory 90 provides volatilestorage for computer software instructions 92 and data 94 used toimplement an embodiment of the present invention (e.g., NRTL-SAC andeNRTL-SAC in FIGS. 3-4). Disk storage 95 provides non-volatile storagefor computer software instructions 92 and data 94 used to implement anembodiment of the present invention. Central processor unit 84 is alsoattached to system bus 79 and provides for the execution of computerinstructions.

In one embodiment, the processor routines 92 and data 94 are a computerprogram product (generally referenced 92 or 20), including a computerreadable medium (e.g., a removable storage medium such as one or moreDVD-ROM's, CD-ROM's, diskettes, tapes, etc.) that provides at least aportion of the software instructions for the invention system 20.Computer program product 92 can be installed by any suitable softwareinstallation procedure, as is well known in the art. In anotherembodiment, at least a portion of the software instructions may also bedownloaded over a cable, communication and/or wireless connection. Inother embodiments, the invention programs are a computer programpropagated signal product 107 embodied on a propagated signal on apropagation medium (e.g., a radio wave, an infrared wave, a laser wave,a sound wave, or an electrical wave propagated over a global networksuch as the Internet, or other network(s)). Such carrier medium orsignals provide at least a portion of the software instructions for thepresent invention routines/program 92.

In alternate embodiments, the propagated signal is an analog carrierwave or digital signal carried on the propagated medium. For example,the propagated signal may be a digitized signal propagated over a globalnetwork (e.g., the Internet), a telecommunications network, or othernetwork. In one embodiment, the propagated signal is a signal that istransmitted over the propagation medium over a period of time, such asthe instructions for a software application sent in packets over anetwork over a period of milliseconds, seconds, minutes, or longer. Inanother embodiment, the computer readable medium of computer programproduct 92 (e.g., NRTL-SAC or eNRTL-SAC) is a propagation medium thatthe computer system 50 may receive and read, such as by receiving thepropagation medium and identifying a propagated signal embodied in thepropagation medium, as described above for computer program propagatedsignal product.

Generally speaking, the term “carrier medium” or transient carrierencompasses the foregoing transient signals, propagated signals,propagated medium, storage medium and the like.

FIGS. 3 and 4 illustrate data flow and process steps for a model 20 (themodel 20 refers to the present invention in context of both NRTL-SAC andeNRTL-SAC models.) performing the methods of the present invention. Withreference to FIG. 3, chemical data describing one or more chemicalspecies (e.g., an electrolyte and solvent) of the mixture and/orenvironmental conditions (e.g., pressure and/or temperature) is enteredat step 105 of the modeler process. Step 110 uses that data to determineat least one conceptual segment including a conceptual segment (fornonelectrolytes) or a conceptual electrolyte segment (for electrolytes)for each of the chemical species of the mixture. The determinedconceptual segment or electrolyte conceptual segment and otherdetermined conceptual segments are used to compute at least one physicalproperty of the mixture during step 115. The computed physicalproperties are analyzed to form a model of at least one physicalproperty of the mixture (e.g., solubility of one or more chemicalspecies in one or more phases of the mixture) in step 120. The modelinformation is then given as output at step 125. The output can take theform of data or an analysis appearing on a computer monitor, data orinstructions sent to a process control system or device, data enteredinto a data storage device, and/or data or instructions relayed toadditional computer systems or programs.

FIGS. 4 a and 4 b illustrate in more detail the computation at step 115in FIG. 3. Step 115 begins with the receipt of determined conceptualelectrolyte and other segments for each of the chemical species (e.g.,nonelectrolytes, electrolyte, solvent, etc.) of the mixture. Thedetermined conceptual segments and the equation:

$\begin{matrix}{{\ln \; \gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}} & (1)\end{matrix}$

are used to compute at least one physical property of the mixture duringstep 215 in FIG. 4 a. As for the determined conceptual electrolyte andother segments shown in FIG. 4 b, the equation:

$\begin{matrix}\begin{matrix}{{\ln \; \gamma_{I}^{*}} = {{\ln \; \gamma_{I}^{*{lc}}} + {\ln \; \gamma_{I}^{*{PDH}}} + {\ln \; \gamma_{I}^{*{FH}}} + {\Delta \; \ln \; \gamma_{I}^{Born}}}} \\{= {{\sum\limits_{m}{r_{m,I}\left( {{\ln \; \Gamma_{m}^{*{lc}}} + {\ln \; \Gamma_{m}^{*{PDH}}}} \right)}} +}} \\{{{r_{c,I}\left( {{\ln \; \Gamma_{c}^{*{lc}}} + {\ln \; \Gamma_{c}^{*{PDH}}} + {{\Delta ln}\; \Gamma_{c}^{Born}}} \right)} +}} \\{{{r_{a,I}\left( {{\ln \; \Gamma_{a}^{*{lc}}} + {\ln \; \Gamma_{a}^{*{PDH}}} + {{\Delta ln}\; \Gamma_{a}^{Born}}} \right)} + {\ln \; \gamma_{I}^{*{FH}}}}}\end{matrix} & (2)\end{matrix}$

are used to compute at least one physical property of the mixture duringstep 215. The computed physical properties are provided as output 220from computation step 215. In step 220, the computed physical propertiesare passed to step 120 of FIG. 16 for forming a model of the physicalproperty of the mixture as described above.

Chromatography Applications of the Present Invention

The term “chemical species” refers to atoms, molecules, molecularfragments, ions, solutes, solvents, and others, as entities beingsubjected to a chemical process or to a measurement.

The term “molecular descriptor” used herein is the result of a logic andmathematical model procedure which transforms intrinsic chemicalinformation encoded within a molecule or expressed as results of somestandardized experiments into certain symbolic representation for thepurpose of correlation, prediction or analysis. One example of“molecular descriptor” is, in addition to the scope and the contentthereof described earlier, conceptual segments, which is based onexpressed characteristics of molecular interactions between chemicalspecies (e.g., a specie in a solute interacting with the surface of thestationary phase.). In certain embodiments, conceptual segments describefor each solute and solvent molecule their effective surface interactionin terms of the following types of the conceptual segments: hydrophobicsegment, polar segment, hydrophilic segment, and solvation segment or acombination thereof. The conceptual segments are to simulate theinteraction characteristics of representative molecular surfaces. Forexample, the hydrophilic segment simulates polar molecular surfaces thatare “hydrogen bond donor or acceptor.” As such, it represents molecularsurfaces with the tendency to form a hydrogen bond. Conversely, thehydrophobic segment simulates molecular surfaces with the adversity toform a hydrogen bond. As for the polar segment, it simulates polarmolecular surfaces that are “electron pair donor or acceptor.” With theconceptual segments identified, real chemical species are then selectedas reference chemical species for the conceptual segments and availablephase equilibrium data of these reference chemical species are used toidentify molecular interaction energy parameters for the conceptualsegments. The reference chemical species with distinct molecularcharacteristics (i.e., hydrophobic, hydrophilic, polar, or solvation)and with abundant, publicly available phase equilibrium data are chosenfor practical purposes.

The methods disclosed herein have chemical and industrial applicationssuch as for manufacture of a drug. One of the applications is in thearea of chromatography, which has become the premier technique forseparations and analyses. Most separation methods in chromatographyemploy two phases, the stationary phase and the mobile phase. The mobilephase can include one or more solvents; and in the mobile phase, one ormore solutes or chemical species are dissolved in the one or moresolvents of the mobile phase to be separated by a chromatographictechnique. The one or more chemical species can include apharmaceutically active ingredient such as a small molecule likepharmaceutical drug molecules (e.g., organic molecules having molecularweight less than about 1000 D, a peptide or a DNA molecule). Conversely,the stationary phase is the part of the chromatographic system throughwhich the mobile phase flows where distribution of the solutes betweenthe phases occurs. The stationary phase may be a solid or a liquid thatis immobilized or adsorbed on a solid.

One embodiment of the present invention related to chromatography ismethods of conducting chromatography for separating one or more chemicalspecies from a mixture. The methods include:

-   -   assigning molecular descriptors based on molecular interaction        characteristics to each of: (i) the one or more species, (ii) a        mobile phase component, and (iii) a stationary phase component;    -   measuring respective molecular interaction indices for the one        or more species, for the mobile phase component and for the        stationary phase component, and    -   determining respective activity coefficients of: (i) the one or        more species; (ii) the mobile phase component; and (iii) the        stationary phase component, such that the determined respective        activity coefficients enables conducting chromatography for        separating the one or more species from the mixture.

In certain embodiments, the method can further include separating theone or more chemical species from the mixture. The word “separation” or“separating” used herein is the hypothetical condition where there iscomplete isolation, by distinct chromatograpic peaks, of each of the xchemical species, which comprises a mixture. It should be understoodthat the adjective “hypothetical” in the definition of “separation” or“separating” above is used because: (1) it is theoretically impossibleto accomplish the complete separation of the species of a mixture;and/or (2) the separated components in the present invention do notrequire to be actually isolated into vessels but rather are detected andtheir presence recorded (e.g., on chart paper or in a computer datafile).

The term “component” used in the “stationary phase component” or “mobilephase component” refers the composition that make up each of the mobilephase and the stationary phase associated with a chromatographyreaction. The composition can include one or more distinct chemicalspecies. For example, the mobile phase component can include one or moresolvents. Conversely, the stationary phase component can include one ormore solvents and/or one or more interacting surface areas. The one ormore interacting surface areas referred herein can be a portion of thesubstance(s), which is fixed in place for the chromatography procedurethat are exposed to the solvents and/or the chemical species in themixture.

The terms “molecular interaction energy indices” refer to measurement ofthe effective surface areas of the chemical species that exhibit surfaceinteraction characteristics. An example of such molecular interactionenergy indices is equivalent numbers. These equivalent numbers are to bedetermined not from the molecular structure of given chemical speciesbut from the interaction characteristics of the chemical species in amixture as expressed in their experimental phase equilibrium data.

In certain embodiments of the present invention, the one or morepredetermined molecular descriptors are based on surface molecularinteraction characteristics. The term “molecular interactioncharacteristics” referred herein are to be determined not from themolecular structure but from the interaction characteristics of thechemical specie in solution as expressed in their experimental phaseequilibrium. Each of these molecular interaction characteristics can berepresented by a conceptual segment as defined above.

The identities of conceptual segments for chemical species can includeat least one conceptual segment (e.g., at least 1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12 or more than 12 conceptual segments) such as hydrophobicsegment, polar segment, hydrophilic segment, solvation segment or acombination thereof. Additional conceptual segments may be introduceddepending on the scope of the different type of chemical speciesinvolved such as organic electrolytes, charged molecules, zwitterions,etc. In certain embodiments, the choice of conceptual segments can be aminimal set rather than a comprehensive set. These conceptual segmentsare chosen to simulate the interaction characteristics of representativemolecular surfaces that significantly contribute to the liquid-phasenonideality of real molecules. For example, the hydrophilic segmentsimulates polar molecular surfaces that are “hydrogen bond donor oracceptor.” As such, it represents molecular surfaces with the tendencyto form a hydrogen bond. The hydrophobic segment simulates molecularsurfaces with the adversity to form a hydrogen bond. The polar segmentsimulates polar molecular surfaces that are “electron pair donor oracceptor.” In certain embodiments, the polar conceptual segment caninclude other non-hydrophobic and non-hydrophilic surface interactions.

Table 1 shows 62 chemical species and their molecular characteristics.As indicated by Table 1, hydrocarbon solvents (aliphatic or aromatic),halogenated hydrocarbons, and ethers are mainly hydrophobic. Ketones,esters, and amides are both hydrophobic and polar. Alcohols, glycols,and amines may have both substantial hydrophilicity and hydrophobicity.Acids are “complex” molecules, exhibiting hydrophilicity, polarity, andhydrophobicity.

In certain embodiments, the step of measuring respective molecularinteraction indices for the one or more chemical species, for the mobilephase component and for the stationary phase component includesdetermining the molecular interaction indices based on measures ofeffective surface areas expressing surface molecular interactioncharacteristics for each of the one or more species, of the mobile phasecomponent, and of the stationary phase component.

In certain embodiments, the step of assigning molecular descriptorsbased on molecular interaction characteristics to each of: (i) the oneor more species, (ii) a mobile phase component, and (iii) a stationaryphase component includes assigning one or more predetermined conceptualsegments to each of the one or more species, of the mobile phasecomponent and of the stationary phase component.

In certain embodiments, the method further includes determiningmolecular interaction energy indices associated with each of thepredetermined conceptual segments.

In certain embodiments, the step of assigning one or more predeterminedconceptual segments to each of the one or more species, of the mobilephase component and of the stationary phase component includesidentifying respective pairwise segment-segment interaction parametersfor each of the predetermined conceptual segments. In certainembodiment, the respective pairwise segment-segment interactionparameters are binary quantities such as binary NRTL parameters. Themethod further defines conceptual segments that broadly characterizesurface interaction characteristics of chemical species, includingmolecules, solvents or solutes. These conceptual segments, together withtheir corresponding nonrandomness factor and segment-segment binaryinteraction energy parameters, are capable of qualitatively describinginteractions between different chemical species, including the varioussolvent-solvent, solvent-solute and solute-solute molecular interactionsand the phase behavior of solvents and solutes. In certain embodiments,the method describes the molecular surface interactions of all solventsand solutes in solution with different types of conceptual segments suchas: hydrophobic segment, solvation segment, polar segment, hydrophilicsegment or a combination thereof. The equivalent numbers for eachchemical species are measurement of the effective molecular surfaceareas that exhibit surface interaction characteristics ofhydrophobicity, solvation, polarity, hydrophilicity or a combinationthereof. For example, hydrophilic segment simulates molecular surfacesthat are “hydrogen bond donor or acceptor”. The hydrophobic segmentsimulates molecular surfaces that show aversion to forming a hydrogenbond. The polar and solvation segments simulate molecular surfaces thatare “electron pair donor or acceptor.” The solvation segment isattractive to hydrophilic segment while the polar segment is repulsiveto hydrophilic segment.

In certain embodiments, the equivalent numbers for each of thepredetermined conceptual segments of the chemical species are determinedin comparison to reference chemical species from regression of availableexperimental equilibrium data associated with these reference chemicalspecies. In certain embodiment, available phase equilibrium data ofthese reference chemical species are used to identify NRTL binaryparameters for the respective conceptual segments. Chemical species withdistinct molecular characteristics (i.e., hydrophobic, hydrophilic, orpolar) and with publicly available phase equilibrium data are chosen asthe reference chemical species.

For example, to determine the conceptual segment numbers of a solutemolecule, solubility data in at least four solvents of varied surfaceinteraction characteristics are chosen. Alternatively, infinite dilutionactivity coefficient data of the solute in at least four solvents ofvaried surface interaction characteristics can be used. Theparameterization is improved if a range of hydrophilic solvents, polarsolvents, solvation solvents and hydrophobic solvents are used. Once thesegment numbers of the solute molecule are determined, the methodsdisclosed herein can then reliably predict solute activity coefficientsand resulting phase behavior in other pure solvents or solvent mixtures.This conceptual segment methodology disclosed herein can also beextended for modeling activity coefficients of other chemical species.

In certain embodiments, the step of assigning one or more predeterminedconceptual segments to each of the one or more species, of the mobilephase component and of the stationary phase component includesdescribing the molecular interactive characteristics in terms of ahydrophobic segment, a polar segment, a solvation segment, a hydrophilicsegment or a combination thereof.

In certain embodiments, measuring respective molecular interactionindices for the one or more chemical species, and the mobile phasecomponent and the stationary phase component includes determining anequivalent number for the predetermined conceptual segment.

In certain embodiments, determining respective activity coefficients of:(i) the one or more species; (ii) the mobile phase component; and (iii)the stationary phase component includes using the following formula:

lnγ_(I)=lnγ_(I) ^(C)+lnγ_(I) ^(R),  (3)

wherein:

-   -   γ_(I) is an activity coefficient for the one or more species,        for the mobile phase component or for the stationary phase        component;    -   γ_(I) ^(C) is a combinatorial contribution to the activity        coefficient for the one or more species, for the mobile phase        component or for the stationary phase component; and    -   γ_(I) ^(R) is a residual contribution to the activity        coefficient of the one or more species, for the mobile phase        component or for the stationary phase component.

In certain embodiment, determining respective activity coefficients of:(i) the one or more species; (ii) the mobile phase component; and (iii)the stationary phase component further includes the following formulasfor computing respective activity coefficients:

$\begin{matrix}{\mspace{79mu} {{{{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {\ln_{m}^{lc} - {\ln \; \Gamma_{m,I}^{lc}}} \right\rbrack}}}},{{\ln \; \Gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}},\text{}\mspace{20mu} {and}}{{{\ln \; \Gamma_{m,I}^{lc}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j,I}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} \right)}}}},}}} & (4)\end{matrix}$

wherein:

-   -   I is a component index for the one or more species, the mobile        phase component, or the stationary phase component;    -   each j, k, m, and m′ is a segment species index;    -   x_(j) is a segment-based mole fraction of segment species j;    -   x_(j,I) is a segment-based mole fraction of segment species j in        I;    -   r_(m,I) is an equivalent number of segment species m contained        only in I;    -   Γ_(m) ^(lc) is an activity coefficient of segment species m;    -   Γ_(m) ^(lc,I) is an activity coefficient of segment species m        contained in I; and    -   G and τ are binary quantities related to each other by a        non-random factor parameter α as in G=exp(−ατ)

In certain embodiments, the method of conducting chromatography forseparating one or more species from a mixture further includescalculating respective retention times of the one or more chemicalspecies based on the respective activity coefficients of: (i) the one ormore species; (ii) the mobile phase component: and (iii) the stationaryphase component. In certain embodiments, calculating respectiveretention times of the one or more chemical species includes computing acapacity ratio for each of the one or more chemical species. Thecapacity ratio is a function of the molecular interaction: a) betweenthe one or more species and a mobile phase component; and b) between theone or more species and a stationary phase component.

In certain embodiments, computing the capacity ratio includes using thefollowing formulas:

$\begin{matrix}{{{k_{I} = \frac{t_{rI} - t_{0}}{t_{0}}},{and}}{{k_{I} = {\frac{x_{s}V_{s}}{x_{m}V_{m}} = {K_{I}\Phi}}},}} & (5)\end{matrix}$

wherein:

-   -   k_(I) is a capacity ratio of the one or more species;    -   t_(rI) is a retention time of the one or more species;    -   t₀ is an elution time for an inert tracer;    -   K_(I) is a partition coefficient;    -   Φ is a phase ratio;    -   V_(s) is a volume of the stationary phase component;    -   V_(m) is a volume of the mobile phase component;    -   x_(s) is a concentration of the one or more species interacting        with the stationary phase component; and    -   x_(m) is a concentration of the one or more species interacting        with the mobile phase component.

In certain embodiment, computing the capacity ratio includes furtherincludes computing the partition coefficient K_(I) using the followingformula:

$\begin{matrix}{{K_{I} = \frac{\gamma_{m}^{\infty}}{\gamma_{s}^{\infty}}},} & (6)\end{matrix}$

wherein:

-   -   γ_(m) ^(∞) and γ_(s) ^(∞) the mobile phase and the stationary        phase activity coefficients of the one or more chemical species        at infinite dilution;

In certain embodiments, the method of conducting chromatography forseparating a one or more species from a mixture further includesselecting independently a composition of the mobile phase component, acomposition of the stationary phase component and a temperature forseparating a desired species from the mixture.

One embodiment of the present invention is a method of conductingchromatography. The method includes:

-   -   controlling a retention time of one or more species in a mixture        by determining at least one conceptual segment of: a) the one or        more species, b) a mobile phase component, and c) a stationary        phase component; and    -   defining an identity and an equivalent number of each of the at        least one conceptual segment.

In certain embodiments, the method further includes separating the oneor more species from the mixture.

In certain embodiments, the one or more chemical species is an activepharmaceutical ingredient.

In certain embodiments, the mobile phase component includes one or moresolvents.

In certain embodiments, the stationary component includes one or moresolvents and interacting surface areas.

In certain embodiments, the at least one conceptual segment is amolecular descriptor to represent a molecular interaction characteristicof; a) the one or more species, b) the mobile phase component, and c)the stationary phase component. In certain embodiments, the molecularinteraction characteristic is a molecular surface interactioncharacteristic of a) the one or more species, b) the mobile phasecomponent, and c) the stationary phase component.

In certain embodiments, the conceptual segment includes a hydrophobicsegment, a polar segment, a hydrophilic segment, or a solvation segment.

In certain embodiments, controlling a retention time of one or morespecies in a mixture includes computing at least one physical propertyfor each of the one or more species, the mobile phase component and thestationary phase component, the at least one physical property computedusing the determined at least one conceptual segment and the equivalentnumber. In certain embodiments, the at least one physical property is anactivity coefficient for each of the one or more species, the mobilephase component and the stationary phase component. In certainembodiments, computing each of the activity coefficient for the one ormore species, for the mobile phase component and for the stationaryphase component includes selecting a plurality of real molecules asreference molecules, each of the reference molecules having at least oneconceptual segment and an equivalent number. In general, the referencemolecules are molecules with: (a) publicly available equilibrium data;and/or (2) particular molecular characters. Molecular characters caninclude hydrophobicity, hydrophilicity, solvation, polarity or acombination thereof. For example, hydrocarbon solvents (aliphatic oraromatic), halogenated hydrocarbons, and ethers are mainly hydrophobic.Conversly, a molecule can exhibit more than one molecular character.Ketones, esters, and amides are both hydrophobic and polar. Alcohols,glycols, and amines may have both substantial hydrophilicity andhydrophobicity. Acids are “complex” molecules, exhibitinghydrophilicity, polarity, and hydrophobicity.

In certain embodiments, computing each of the activity coefficient forinteraction of the one or more chemical species, the mobile phasecomponent and the stationary phase component further includesidentifying the equivalent numbers for the one or more species, for themobile phase component, and for the stationary phase component, withrespect to the reference molecules.

In certain embodiments, computing each of the activity coefficient forthe one or more chemical species, the mobile phase component and thestationary phase component includes providing data regarding partitioncharacteristics of the reference molecules in a chromatographyseparation. In certain embodiments, computing each of the activitycoefficient for the one or more chemical species, the mobile phasecomponent and the stationary phase component further includesdetermining partition characteristics of the one or more species withrespect to the partition characteristics of the reference molecules. Thedata regarding partition characteristics of the reference molecules in achromatography separation referred in these embodiments is partitioncoefficient data or retention factor data.

In certain embodiments, the method of conducting chromatography furtherincludes determining molecular interaction energy indices associatedwith each of the conceptual segments. As previously addressed, anexample of the molecular interaction energy indices is binary NRTLparameters.

In certain embodiments, the method of conducting chromatography furtherincludes further computing activity coefficients of the one or morespecies, the mobile phase component and the stationary phase component.

In certain embodiments, computing the activity coefficient of the one ormore species, the mobile phase component or the stationary phasecomponent uses the following formulae:

lnγ_(I)=lnγ_(I) ^(C)+lnγ_(I) ^(R),  (7)

wherein:

-   -   γ_(I) is an activity coefficient for the one or more species,        the mobile phase component or the stationary phase component;    -   γ_(I) ^(C) is a combinatorial contribution to the activity        coefficient for the one or more species, the mobile phase        component or the stationary phase component; and    -   γ_(I) ^(R) is a residual contribution to the activity        coefficient of the one or more species, the mobile phase        component or the stationary component.

In certain embodiments, computing the activity coefficient of the one ormore species, the mobile phase component or the stationary phasecomponent includes computing the following formulas:

$\begin{matrix}{\mspace{79mu} {{{{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m,I}^{lc}}} \right\rbrack}}}},{{\ln \; \Gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}},\mspace{20mu} {and}}{{{\ln \; \Gamma_{m,I}^{lc}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j,I}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} \right)}}}},}}} & (8)\end{matrix}$

wherein:

-   -   I is a component index for the one or more species, the mobile        phase component, or the stationary phase component;    -   each j, k, m, and m′ is a segment species index;    -   x_(j) is a segment-based mole fraction of segment species j;    -   x_(j,1) is a segment-based mole fraction of segment species j in        the I;    -   r_(m,I) is a number of segment species m contained only in the        I;    -   Γ_(m) ^(lc) is an activity coefficient of segment species m;    -   Γ_(m) ^(lc,I) is an activity coefficient of segment species m        contained in the I; and    -   G and τ are binary quantities related to each other by a        non-random factor parameter α as in G=exp(−ατ)

In certain embodiments, controlling a retention time of the one or morespecies includes calculating the retention time of the one or morespecies. In certain embodiments, calculating the retention time of theone or more species includes computing a capacity ratio of the one ormore species. A certain embodiment of the present invention is acomputer program product.

The computer program product includes:

-   -   a) a computer usable medium; and    -   b) a set of computer program instructions embodied on the        computer usable medium for conducting chromatography for        separating a one or more chemical species from a mixture by:        -   1) assigning molecular descriptors based on molecular            interaction characteristics to each of: (i) the one or more            species, (ii) a mobile phase component, and (iii) a            stationary phase component;        -   2) measuring respective molecular interaction indices for            the one or more species, for the mobile phase component and            for the stationary phase component; and        -   3) determining respective activity coefficients of: (i) the            one or more species; (ii) the mobile phase component;            and (iii) the stationary phase component, such that the            determined respective activity coefficients enables            conducting chromatography for separating the one or more            species from the mixture.

In certain embodiments, at least some portion of the computer programcan be transmitted over a global network. In certain embodiments, thecomputer usable medium can include a removable storage medium. Theremoval storage medium can include a CD-ROM, a DVD-ROM, a diskette, atape or a combination thereof.

In certain embodiments, the one or more chemical species includes anactive pharmaceutical ingredient.

In certain embodiments, the set of computer program instructionsembodied on the computer usable medium for conducting chromatography forseparating one or more chemical species from a mixture can include anadditional step of computing at least one physical property for each ofthe one or more species, for the mobile phase component and for thestationary phase component using the determined at least one conceptualsegment.

In certain embodiments, the set of computer program instructionsembodied on the computer usable medium for conducting chromatography forseparating one or more chemical species from a mixture can include anadditional step of selecting independently a solvent composition of themobile phase component, the stationary phase component and a temperaturefor separating a desired species from the mixture.

A certain embodiment of the present invention is a computer system forconducting chromatography. The computer system for conductingchromatography includes:

-   -   a) a user input means for obtaining empirical data from a user;    -   b) a digital processor coupled to receive the empirical data        from the input means, wherein the digital processor executes a        modeling system in working memory, wherein the modeling system        uses the empirical data for:        -   1) assigning molecular descriptors based on molecular            interaction characteristics to each of: (i) each of the one            or more species; (ii) a mobile phase component; and (iii) a            stationary phase component;        -   2) measuring respective molecular interactive indices for            the one or more species, and the mobile phase component and            for the stationary phase component; and        -   3) determining activity coefficients of: (i) the one or more            species; (ii) the mobile phase component; and (iii) the            stationary phase component such that the determined            respective activity coefficients enables conducting            chromatography for separating the one or more species from            the mixture.

In a certain embodiment, the computer system for conductingchromatography can further include an output means coupled to thedigital processor. The output means provides to the user indications ofselecting the mobile phase, and the stationary phase and, optionally, areaction temperature for the one or more species from the modelingsystem.

In certain embodiments, the computer system enables transmission of someportion of the empirical data and results from the modeling system overa global network.

It should be understood that the method of conducting chromatographydisclosed herein can be applied to any type of chromatographictechniques. Examples of techniques include any one of: gas-liquid,gas-solid, supercritical fluid, liquid-liquid, thin layer, liquid solid,ion-exchange, size-excursion, affinity, bonded phase, capillaryelectrochromatography, or capillary zone electrochromatography.

Yet in certain embodiments, the present invention is a method ofdetermining the conceptual segment numbers for a chromatographictechnique where a solute is an electrolyte. The segment number of anelectrolyte accounts for additional segment numbers that are not presentwith a non-electrolyte species. For example, the segment interactionconcept provides the framework to account for the attractive interactionions with the hydrophilic segments of organic solvents and the repulsiveinteraction of ions with the hydrophobic segments of organic solvents.Once the segment numbers of the solute are determined, the methodsdisclosed herein can then predict solute activity coefficients andresulting phase behavior in one or more solvents. In a certainembodiment, the electrolyte is an active pharmaceutical ingredient.

One embodiment of the present invention is a method of determiningequivalent numbers of a stationary phase component for a chromatographyanalysis. The conceptual segment numbers of the stationary phasecomponent can be determined from measuring a degree of surfaceinteraction of the plurality of predetermined solutes with thestationary phase when each of the plurality of predetermined solutesinteracts with the stationary phase. The degree of the surfaceinteractions between the plurality of predetermined solutes and thestationary phase can be measured by obtaining partition coefficient dataor retention factor data of the plurality of the predetermined solutes.Such partition coefficient data or retention factor data reflects thenature of the solute-stationary phase surface interaction that is abasis for the determination of the conceptual segment numbers for themakeup of the stationary phase. The types of the solute-stationary phasesurface interaction can include hydrophobic interaction, electrostaticsolvation interaction, electrostatic polar interaction, and hydrophilicinteraction.

The methods disclosed herein have many practical applications formodeling chromatography reactions. For example, if the solubilities ofthe chemical species are known, the model parameters for the chemicalspecies could be established by their solubilities in the same way thatthe parameters are found when the model is used to predict solubilities.However, when a chemical species is being purified chromatographically,it often has not yet reached a stage of development where sufficientamounts of it are available to measure its solubility in severalsolvents. Furthermore, the chromatographic methods disclosed herein arecapable of predicting the retention times of impurities as well as theretention time for the chemical species so that the chemical species maybe separated from the impurities based on a difference in the retentiontimes. For the methods disclosed herein to predict the retention timesfor the impurities, parameters for the impurities can be established,which could be done based on the solubilities of the impurities.

The stationary phase can be regarded as another solvent, the parametersfor the model for determining retention time of a chemical species canbe established based on the parameters that had already been establishedwith the stationary phase component. This parameter can describe theinteractions between a molecule of a given size of the component and thevoids through which it can move which depend on the size of thestationary phase substrate. In order to determine this parameter, one ortwo chromatographic measurements of retention time can be done persolute on a reference column and used as input to the model indetermining this parameter.

Other parameters such as the size of the substrate that makes up thestationary phase component can be added to enhance the quality of themodel.

The following Examples are illustrative of the invention, including theembodiment that are related to chromatography applications and are notmeant to be limiting in any way.

Example 1 Modeling a Mixture of Nonelectrolyte Chemical Species

A study was performed to determine how well the NRTL-SAC models thesolubility of mixtures comprising a solid organic nonelectrolyte.

The solubility of a solid organic nonelectrolyte is described well bythe expression:

$\begin{matrix}{{\ln \; x_{I}^{SAT}} = {{\frac{\Delta_{fus}S}{R}\left( {1 - \frac{T_{m}}{T}} \right)} - {\ln \; \gamma_{I}^{SAT}}}} & (9)\end{matrix}$

for T≦T_(m) and where the entropy of fusion of the solid (Δ_(fus)S) isrepresented by:

$\begin{matrix}{{\Delta_{fus}S} = \frac{\Delta_{fus}H}{T_{m}}} & (10)\end{matrix}$

X_(I) ^(SAT) is the mole fraction of the solid (the solute) dissolved inthe solvent phase at saturation, γ_(I) ^(SAT) is the activitycoefficient for the solute in the solution at saturation, R is the gasconstant, T is the temperature, and T_(m) is the melting point of thesolid. Given a polymorph, Δ_(fus)S and T_(m) are fixed and thesolubility is then a function of temperature and activity coefficient ofthe solute in the solution. The activity coefficient of the solute inthe solution plays the key role in determining the solubility. Ingeneral, the activity coefficient of the solute in the solution isusually calculated from a liquid activity coefficient model.

Except for the ideal solution model, an activity coefficient model isoften written in two parts as such:

lnγ_(I)=lnγ_(I) ^(C)+lnγ_(I) ^(R)  (11)

γ_(I) ^(C) and γ_(I) ^(R) are the combinatorial and residualcontributions to the activity coefficient of component I, respectively.

In NRTL-SAC, the combinatorial part, γ_(I) ^(C), is calculated from theFlory-Huggins term for the entropy of mixing. The residual part, γ_(I)^(R), is set equal to the local composition (lc) interactioncontribution, γ_(I) ^(lc):

$\begin{matrix}{\mspace{79mu} {{{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {{\ln \; \gamma_{m}^{lc}} - {\ln \; \gamma_{m}^{{lc},I}}} \right\rbrack}}}}\mspace{20mu} {with}}} & (12) \\{{{\ln \; \gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}},{{\ln \; \gamma_{m}^{{lc},I}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} \right)}}}},\mspace{20mu} {x_{j} = \frac{\sum\limits_{J}{x_{J}r_{j,J}}}{\sum\limits_{I}{\sum\limits_{i}{x_{I}r_{i,I}}}}},\mspace{20mu} {x_{j,I} = \frac{r_{j,I}}{\sum\limits_{j}r_{j,I}}},} & (13)\end{matrix}$

where i,j,k,m,m′ are the segment-based species index, I, J are thecomponent index, x_(j) is the segment-based mole fraction of segmentspecies j, and x_(J) is the mole fraction of component J, r_(m,I), isthe number of segment species m contained in component I, γ_(m) ^(lc) isthe activity coefficient of segment species m, and γ_(m) ^(lc,I) is theactivity coefficient of segment species m contained only in component I.G and τ are local binary quantities related to each other by the NRTLnon-random factor parameter α:

G=exp(−ατ)  (14)

The equation:

$\begin{matrix}{{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {{\ln \; \gamma_{m}^{lc}} - {\ln \; \gamma_{m}^{{lc},I}}} \right\rbrack}}}} & (15)\end{matrix}$

is a general form for the local composition interaction contribution toactivity coefficients of components in the NRTL-SAC model of the presentinvention. For mono-segment solvent components(S), this equation can besimplified and reduced to the classical NRTL model as follows:

$\begin{matrix}{\mspace{79mu} {{{\ln \; \gamma_{I = S}^{lc}} = {\sum\limits_{m}{r_{m,S}\left\lbrack {{\ln \; \gamma_{m}^{lc}} - {\ln \; \gamma_{m}^{{lc},S}}} \right\rbrack}}}\mspace{20mu} {with}}} & (16) \\{\mspace{79mu} {{{r_{m,S} = 1},{{\ln \; \gamma_{m}^{{lc},S}} = 0.}}\mspace{20mu} {{Therefore},}}} & (17) \\{{{\ln \; \gamma_{I = S}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jS}\tau_{jS}}}{\sum\limits_{k}{x_{k}G_{kS}}} + {\sum\limits_{m}{\frac{x_{m}G_{Sm}}{\sum\limits_{k}{x_{k}G_{km}}}\left( {\tau_{Sm} - \frac{\sum\limits_{k}{x_{k}G_{km}\tau_{km}}}{\sum\limits_{k}{x_{k}G_{km}}}} \right)}}}},\mspace{20mu} {where}} & (18) \\{\mspace{79mu} {{G_{jS} = {\exp \left( {{- \alpha_{jS}}\tau_{jS}} \right)}},{G_{Sj} = {{\exp \left( {{- \alpha_{jS}}\tau_{Sj}} \right)}.}}}} & (19)\end{matrix}$

This is the same equation as the classical NRTL model.

Three conceptual segments were defined for nonelectrolyte molecules: ahydrophobic segment, a polar segment, a hydrophilic segment and asolvation segment. These conceptual segments qualitatively capture thephase behavior of real molecules and their corresponding segments. Realmolecules in turn are used as reference molecules for the conceptualsegments and available phase equilibrium data of these referencemolecules are used to identify NRTL binary parameters for the conceptualsegments. Preferably, these reference molecules possess distinctmolecular characteristics (i.e., hydrophobic, hydrophilic, or polar) andhave abundant, publicly available, thermodynamic data (e.g., phaseequilibrium data). The study was focused on the 59 ICH solvents used inpharmaceutical process design. Water, triethylamine, and n-octanol werealso considered. Table 1 shows these 62 solvents and the solventcharacteristics.

TABLE 1 Common Solvents in Pharmaceutical Process Design Solvent Solvent(Component 1) τ₁₂ ^(a) τ₂₁ ^(a) τ₁₂ ^(b) τ₂₁ ^(b) τ₁₂ ^(c) τ₂₁ ^(c)characteristics ACETIC-ACID 1.365 0.797 2.445 −1.108 Complex ACETONE0.880 0.935 0.806 1.244 Polar ACETONITRILE 1.834 1.643 0.707 1.787 PolarANISOLE Hydrophobic BENZENE 1.490 −0.614 3.692 5.977 Hydrophobic1-BUTANOL −0.113 2.639 0.269 2.870 −2.157 5.843 Hydrophobic/ Hydrophilic2-BUTANOL −0.165 2.149 −0.168 3.021 −1.539 5.083 Hydrophobic/Hydrophilic N-BUTYL-ACETATE 1.430 2.131 Hydrophobic/PolarMETHYL-TERT-BUTYL- −0.148 0.368 1.534 4.263 Hydrophobic ETHERCARBON-TETRACHLORIDE 1.309 −0.850 5.314 7.369 Hydrophobic CHLOROBENZENE0.884 −0.194 4.013 7.026 Hydrophobic CHLOROFORM 1.121 −0.424 3.587 4.954Hydrophobic CUMENE Hydrophobic CYCLOHEXANE −0.824 1.054 6.012 9.519Hydrophobic 1,2-DICHLOROETHANE 1.576 −0.138 3.207 4.284 2.833 4.783Hydrophobic 1,1-DICHLOROETHYLENE Hydrophobic 1,2-DICHLOROETHYLENEHydrophobic DICHLOROMETHANE 0.589 0.325 1.983 3.828 Polar1,2-DIMETHOXYETHANE 0.450 1.952 Polar N,N-DIMETHYLACETAMIDE −0.564 1.109Polar N,N-DIMETHYLFORMAMIDE 1.245 1.636 −1.167 2.044 PolarDIMETHYL-SULFOXIDE −2.139 0.955 Polar 1,4-DIOXANE 1.246 0.097 1.0031.010 Polar ETHANOL 0.533 2.192 −0.024 1.597 Hydrophobic/ Hydrophilic2-ETHOXYETHANOL −0.319 2.560 −1.593 1.853 Hydrophobic/ HydrophilicETHYL-ACETATE 0.771 0.190 0.508 3.828 Hydrophobic/Polar ETHYLENE-GLYCOL1.380 −1.660 Hydrophilic DIETHYL-ETHER −0.940 1.400 1.612 3.103Hydrophobic ETHYL-FORMATE Polar FORMAMIDE Complex FORMIC-ACID −0.340−1.202 Complex N-HEPTANE −0.414 0.398 Hydrophobic N-HEXANE 6.547 10.9496.547 10.949 Hydrophobic ISOBUTYL-ACETATE Polar ISOPROPYL-ACETATE PolarMETHANOL 1.478 1.155 0.103 0.396 Hydrophobic/ Hydrophilic2-METHOXYETHANOL 1.389 −0.566 Hydrophobic/ Hydrophilic METHYL-ACETATE0.715 2.751 Polar 3-METHYL-1-BUTANOL 0.062 2.374 −0.042 3.029 −0.5985.680 Hydrophobic/ Hydrophilic METHYL-BUTYL-KETONE Hydrophobic/PolarMETHYLCYCLOHEXANE 1.412 −1.054 Polar METHYL-ETHYL-KETONE −0.036 1.2730.823 2.128 −0.769 3.883 Hydrophobic/Polar METHYL-ISOBUTYL-KETONE 0.9774.868 Hydrophobic/Polar ISOBUTANOL 0.021 2.027 0.592 2.702 −1.479 5.269Hydrophobic/ Hydrophilic N-METHYL-2-PYRROLIDONE −0.583 3.270 −0.2350.437 Hydrophobic NITROMETHANE 1.968 2.556 Polar N-PENTANE 0.496 −0.523Hydrophobic 1-PENTANOL −0.320 2.567 −0.029 3.583 Hydrophobic/Hydrophilic 1-PROPANOL 0.049 2.558 0.197 2.541 Hydrophobic/ HydrophilicISOPROPYL-ALCOHOL 0.657 1.099 0.079 2.032 Hydrophobic/ HydrophilicN-PROPYL-ACETATE 1.409 2.571 Hydrophobic/Polar PYRIDINE −0.665 1.664−0.990 3.146 Polar SULFOLANE 1.045 0.396 Polar TETRAHYDROFURAN 0.6311.981 1.773 0.563 Polar 1,2,3,4- 1.134 −0.631 HydrophobicTETRAHYDRONAPHTHALENE TOLUENE −0.869 1.292 4.241 7.224 Hydrophobic1,1,1-TRICHLOROETHANE 0.535 −0.197 Hydrophobic TRICHLOROETHYLENE 1.026−0.560 Hydrophobic M-XYLENE Hydrophobic WATER 10.949 6.547 HydrophilicTRIETHYLAMINE −0.908 1.285 1.200 1.763 −0.169 4.997 Hydrophobic/Polar1-OCTANOL −0.888 3.153 0.301 8.939 Hydrophobic/ Hydrophilic Wherein: 1.τ₁₂ ^(a) and τ₂₁ ^(a) are NRTL binary τ parameters for systems of thelisted solvents and hexane. NRTL non-random factor parameter, α, isfixed as a constant of 0.2. In these binary systems, solvent iscomponent 1 and hexane component 2. τ's were determined from availableVLE & LLE data. 2. τ₁₂ ^(b) and τ₂₁ ^(b) are NRTL binary τ parametersfor systems of the listed solvents and water. NRTL non-random factorparameter, α, is fixed as a constant of 0.3. In these binary systems,solvent is component 1 and water component 2. τ's were determined fromavailable VLE data. 3. τ₁₂ ^(c) and τ₂₁ ^(c) are NRTL binary τparameters for systems of the listed solvents and water. NRTL non-randomfactor parameter, α, is fixed as a constant of 0.2. In these binarysystems, solvent is component 1 and water component 2. τ's weredetermined from available LLE data.

Hydrocarbon solvents (aliphatic or aromatic), halogenated hydrocarbons,and ethers are mainly hydrophobic. Ketones, esters and amides are bothhydrophobic and polar. Alcohols, glycols, and amines may have bothsubstantial hydrophilicity and hydrophobicity. Acids are complex, withhydrophilicity, polarity, and hydrophobicity.

Also shown in Table 1 are the available NRTL binary parameters (τ) forvarious solvent-water binary systems and solvent-hexane binary systems.Applicants obtained these binary parameters from fitting selectedliterature phase equilibrium data and deliberately ignoring thetemperature dependency of these parameters. These values illustrate therange of values for these binary parameters. Note that many of thebinary parameters are missing, as the phase equilibrium data is notfound in the literature or simply has never been determined for thatsolvent mixture. Also note the sheer number of binary parameters neededfor the prior art NRTL models for even a moderately sized system ofsolvents. For example, to model 60 solvents with the NRTL model, 60×60NRTL binary parameters would be needed.

Table 1 shows that, for the NRTL binary parameters determined from VLEand LLE data for hydrophobic solvent (1) water (2) binaries, allhydrophobic solvents exhibit similar repulsive interactions with waterand both τ₁₂ and τ₂₁ are large positive values for the solvent-waterbinaries. When the hydrophobic solvents also carry significanthydrophilic or polar characteristics, τ₁₂ becomes negative while τ₂₁retain a large positive value.

Table 1 also illustrates that similar repulsive, but weaker,interactions between a polar solvent (1) and hexane (2), arepresentative hydrophobic solvent. Both τ₁₂ and τ₂₁ are small, positivevalues for the solvent-hexane binaries. The interactions betweenhydrophobic solvents and hexane are weak and the corresponding NRTLbinary parameters are around or less than unity, characteristic ofnearly ideal solutions.

The interactions between polar solvents (1) and water (2) are moresubtle. While all τ₂₁ are positive, τ₁₂ can be positive or negative.This is probably due to different polar molecules exhibiting differentinteractions, some repulsive and others attractive, with hydrophilicmolecules.

Hexane and water were chosen as the reference molecule for hydrophobicsegment and for hydrophilic segment, respectively. The selection ofreference molecule for polar segment requires attention to the widevariations of interactions between polar molecules and water.Acetonitrile was chosen as the reference molecule for a polar segment,and a mechanism was introduced to tune the way the polar segment ischaracterized. The tuning mechanism, as shown in Table 2, allows tuningof the interaction characteristics between the polar segment and thehydrophilic segment. In other words, instead of using only one polarsegment (“Y”), two polar segments (“Y−” and “Y+”) were used. Thedifference between Y− and Y+ is the way they interact with thehydrophilic segment.

The chosen values for the NRTL binary interactions parameters, α and τ,for the three conceptual segments are summarized in Table 2.

TABLE 2 NRTL Binary Parameters for Conceptual Segments in NRTL-SACSegment (1) X (hydro- X (hydro- X (hydro- phobic phobic Y− (polar Y+(polar phobic segment) segment) segment) segment) segment) Segment (2) ZZ (hydro- Z (hydro- Y− (polar (hydrophilic philic philic Y+ (polarsegment) segment) segment) segment) segment) τ₁₂ 1.643 6.547 −2.0002.000 1.643 τ₂₁ 1.834 10.949 1.787 1.787 1.834 α₁₂ = α₂₁ 0.2 0.2 0.3 0.30.2

As a first approximation, the temperature dependency of the binaryparameters was ignored.

The binary parameters for the hydrophobic segment (1)—hydrophilicsegment (2) were determined from available liquid-liquid equilibriumdata of hexane-water binary mixture (see Table 1). α was fixed at 0.2because it is the customary value for a for systems that exhibitliquid-liquid separation. Here both τ₁₂ and τ₂₁ are large positivevalues (6.547, 10.950). They highlight the strong repulsive nature ofthe interactions between the hydrophobic segment and the hydrophilicsegment. Determining a suitable value for α is known in the art. See J.M. PRAUSNITZ, ET AL., MOLECULAR THERMODYNAMICS OF FLUID-PHASE EQUILIBRIA261 (3d ed. 1999).

The binary parameters for the hydrophobic segment (1)—polar segment (2)were determined from available liquid-liquid equilibrium data ofhexane—acetonitrile binary mixture (see Table 1). Again, a was fixed at0.2. Both τ₁₂ and τ₂₁ were small positive values (1.643, 1.834). Theyhighlight the weak repulsive nature of the interactions betweenhydrophobic segment and polar segment.

The binary parameters for the hydrophilic segment (1)—polar segment (2)were determined from available vapor-liquid equilibrium data ofwater—acetonitrile binary mixture (see Table 1). α was fixed at 0.3 forthe hydrophilic segment—polar segment pair because this binary does notexhibit liquid-liquid separation. τ₁₂ was fixed at a positive value(1.787) and r₂₁ was allowed to vary between −2 and 2. Two types of polarsegments were allowed, Y− and Y+. For Y− polar segment, the values ofτ₁₂ and r₂₁ were (1.787, −2). For Y+ polar segment, they were (1.787,2). Note that both Y− polar segment and Y+ polar segment exhibited thesame repulsive interactions with hydrophobic segments as discussed inthe previous paragraph. Also, ideal solution was assumed for Y− polarsegment and Y+ polar segment mixtures (i.e., τ₁₂=τ₂₁=0).

Table 2 captures the general trends for the NRTL binary parameters thatwere observed for a wide variety of hydrophobic, polar, and hydrophilicmolecules.

The application of the NRTL-SAC model requires a databank of molecularparameters for common solvents used in the industry. In this example,each solvent was described by using up to four molecular parameters,i.e., X, Y+, Y−, and Z. So, using four molecular parameters to model asystem of 60 solvents, a set of up to 4×60 molecular parameters would beused. However, due to the fact that these molecular parameters representcertain unique molecular characteristics, often only one or twomolecular parameters are needed for most solvents. For example, alkanesare hydrophobic and they are well represented with hydrophobicity, X,alone. Alcohols are hybrids of hydrophobic segments and hydrophilicsegments and they are well represented with X and Z. Ketones, esters,and ethers are polar molecules with varying degrees of hydrophobiccontents. They are well represented by X and Y's. Hence, the needed setof molecular parameters can be much smaller than 4×60.

Determination of solvent molecular parameters involves regression ofexperimental VLE or LLE data for binary systems of interested solventand the above-mentioned reference molecules (i.e., hexane, acetonitrile,and water) or their substitutes. Solvent molecular parameters are theadjustable parameters in the regression. If binary data is lacking forthe solvent with the reference molecules, data for other binaries may beused as long as the molecular parameters for the substitute referencemolecules are already identified. In a way, these reference moleculescan be thought of as molecular probes that are used to elucidate theinteraction characteristics of the solvent molecules. These molecularprobes express the interactions in terms of binary phase equilibriumdata.

Table 3 lists the molecular parameters identified for the commonsolvents in the ICH list.

TABLE 3 Molecular Parameters for Common Solvents. Solvent name X Y− Y+ ZACETIC-ACID 0.045 0.164 0.157 0.217 ACETONE 0.131 0.109 0.513ACETONITRILE 0.018 0.131 0.883 ANISOLE 0.722 BENZENE 0.607 0.1901-BUTANOL 0.414 0.007 0.485 2-BUTANOL 0.335 0.082 0.355 N-BUTYL-ACETATE0.317 0.030 0.330 METHYL-TERT-BUTYL-ETHER 1.040 0.219 0.172CARBON-TETRACHLORIDE 0.718 0.141 CHLOROBENZENE 0.710 0.424 CHLOROFORM0.278 0.039 CUMENE 1.208 0.541 CYCLOHEXANE 0.892 1,2-DICHLOROETHANE0.394 0.691 1,1-DICHLOROETHYLENE 0.529 0.208 1,2-DICHLOROETHYLENE 0.1880.832 DICHLOROMETHANE 0.321 1.262 1,2-DIMETHOXYETHANE 0.081 0.194 0.858N,N-DIMETHYLACETAMIDE 0.067 0.030 0.157 N,N-DIMETHYLFORMAMIDE 0.0730.564 0.372 DIMETHYL-SULFOXIDE 0.532 2.890 1,4-DIOXANE 0.154 0.086 0.401ETHANOL 0.256 0.081 0.507 2-ETHOXYETHANOL 0.071 0.318 0.237ETHYL-ACETATE 0.322 0.049 0.421 ETHYLENE-GLYCOL 0.141 0.338DIETHYL-ETHER 0.448 0.041 0.165 ETHYL-FORMATE 0.257 0.280 FORMAMIDE0.089 0.341 0.252 FORMIC-ACID 0.707 2.470 N-HEPTANE 1.340 N-HEXANE 1.000ISOBUTYL-ACETATE 1.660 0.108 ISOPROPYL-ACETATE 0.552 0.154 0.498METHANOL 0.088 0.149 0.027 0.562 2-METHOXYETHANOL 0.052 0.043 0.2510.560 METHYL-ACETATE 0.236 0.337 3-METHYL-1-BUTANOL 0.419 0.538 0.314METHYL-BUTYL-KETONE 0.673 0.224 0.469 METHYLCYCLOHEXANE 1.162 0.251METHYL-ETHYL-KETONE 0.247 0.036 0.480 METHYL-ISOBUTYL-KETONE 0.673 0.2240.469 ISOBUTANOL 0.566 0.067 0.485 N-METHYL-2-PYRROLIDONE 0.197 0.3220.305 NITROMETHANE 0.025 1.216 N-PENTANE 0.898 1-PENTANOL 0.474 0.2230.426 0.248 1-PROPANOL 0.375 0.030 0.511 ISOPROPYL-ALCOHOL 0.351 0.0700.003 0.353 N-PROPYL-ACETATE 0.514 0.134 0.587 PYRIDINE 0.205 0.1350.174 SULFOLANE 0.210 0.457 TETRAHYDROFURAN 0.235 0.040 0.3201,2,3,4-TETRAHYDRONAPHTHALENE 0.443 0.555 TOLUENE 0.604 0.3041,1,1-TRICHLOROETHANE 0.548 0.287 TRICHLOROETHYLENE 0.426 0.285 M-XYLENE0.758 0.021 0.316 WATER 1.000 TRIETHYLAMINE 0.557 0.105 1-OCTANOL 0.7660.032 0.624 0.335

Among the ICH solvents, the molecular parameters identified for anisole,cumene, 1,2-dichloroethylene, 1,2-dimethoxyethane,N,N-dimethylacetamide, dimethyl sulfoxide, ethyl formate, isobutylacetate, isopropyl acetate, methyl-butyl-ketone, tetralin, andtrichloroethylene were questionable, due to lack of sufficientexperimental binary phase equilibrium data. In fact, no public data formethyl-butyl-ketone (2-hexanone) was found and its molecular parameterswere set to be the same as those for methyl-isobutyl-ketone.

The NRTL-SAC model with the molecular parameters qualitatively capturesthe interaction characteristics of the solvent mixtures and theresulting phase equilibrium behavior. FIGS. 5 to 7 contain three graphsillustrating the binary phase diagrams for a water, 1,4-dioxane, andoctanol system at atmospheric pressure. The graphs illustrate thepredictions of both the NRTL model with the binary parameters in Table 1and NRTL-SAC models with the model parameters of Table 3. FIG. 5illustrates the water, 1,4-dioxane mixture; FIG. 6 illustrates thewater, octanol mixture; and FIG. 7 illustrates the octanol, 1,4-dioxanemixture. The predictions with the NRTL-SAC model are broadly consistentwith the calculations from the NRTL model that are generally understoodto represent experimental data within engineering accuracy.

Example 2 Model Prediction Results

Data compiled by Marrero and Abildsko provides a good source ofsolubility data for large, complex chemicals. Marrero, J. & Abildskov,J., Solubility and Related Properties of Large Complex Chemicals, Part1: Organic Solutes Ranging from C ₄ to C ₄₀, CHEMISTRY DATA SERIES XV,DECHEMA, (2003). From that applicants extracted solubility data for the8 molecules reported by Lin and Nash. Lin, H.-M. & R. A. Nash, AnExperimental Method for Determining the Hildebrand Solubility Parameterof Organic Electrolytes, 82 J. PHARMACEUTICAL SCI. 1018 (1993). Alsotested, were 6 additional molecules with sizable solubility data sets.

The NRTL-SAC model was applied to the solvents that are included inTable 3.

The molecular parameters determined for the solutes are summarized inTable 4. During the data regression, all experimental solubility data,regardless of the order of magnitude, were assigned with a standarddeviation of 20%. The comparisons between the experimental solubilityand the calculated solubility are given in FIGS. 8 to 21, whichillustrate phase diagrams for the systems at 298.15K and atmosphericpressure.

Good representations for the solubility data was obtained with theNRTL-SAC model. The RMS errors in ln x for the fits are given in Table4.

TABLE 4 Molecular parameters for solutes. RMS # of error on Solute MWsolvents T (K) X Y− Y+ Z lnK_(sp) ln x p-Aminobenzoic 137.14 7 298.150.218 0.681 1.935 0.760 −2.861 0.284 acid Benzoic acid 122.12 7 298.150.524 0.089 0.450 0.405 −1.540 0.160 Camphor 152.23 7 298.15 0.604 0.1240.478 0.000 −0.593 0.092 Ephedrine 165.23 7 298.15 0.458 0.068 0.0000.193 −0.296 0.067 Lidocaine 234.33 7 298.15 0.698 0.596 0.293 0.172−0.978 0.027 Methylparaben 152.14 7 298.15 0.479 0.484 1.218 0.683−2.103 0.120 Testosterone 288.41 7 298.15 1.051 0.771 0.233 0.669 −3.7970.334 Theophylline 180.18 7 298.15 0.000 0.757 1.208 0.341 −6.110 0.661Estriol 288.38  9^(a) 298.15 0.853 0.000 0.291 1.928 −7.652 0.608Estrone 270.37 12  298.15 0.499 0.679 1.521 0.196 −6.531 0.519 Morphine285.34 6 308.15 0.773 0.000 0.000 1.811 −4.658 1.007 Piroxicam 331.3514^(b ) 298.15 0.665 0.000 1.803 0.169 −7.656 0.665 Hydrocortisone362.46 11^(c ) 298.15 0.401 0.970 1.248 0.611 −6.697 0.334 Haloperidol375.86 13^(d ) 298.15 0.827 0.000 0.000 0.131 −4.398 0.311 ^(a)With THFexcluded. ^(b)With 1,2 dichloroethane, chloroform, diethyl ether, andDMF excluded. ^(c)With hexane excluded. ^(d)With chloroform and DMFexcluded.K_(sp), the solubility product constant, corresponds to the idealsolubility (in mole fraction) for the solute. The quality of the fitreflects both the effectiveness of the NRTL-SAC model and the quality ofthe molecular parameters identified from the limited availableexperimental data for the solvents.

FIGS. 8, 9, 10, 11, 12, 13, 14 and 15 include graphs illustrating theexperimental solubilities vs. calculated solubilities for p-aminobenzoicacid, benzoic acid, camphor, ephedrine, lidocaine, methylparaben,testosterone, and theophylline, respectively, in various solvents at298.15K. The various solvents used were selected from a group of 33solvents, including acetic acid, acetone, benzene, 1-butanol, n-butylacetate, carbon tetrachloride, chlorobenzene, chloroform, cyclohexane,1,2-dichloroethane, dichloromethane, 1,2-dimethoxyethane,N,N-dimethylformamide, dimethyl-sulfoxide, 1,4-dioxane, ethanol,2-ethoxyethanol, ethyl acetate, ethylene glycol, diethyl ether,formamide, n-heptane, n-hexane, isopropyl acetate, methanol, methylacetate, 1-pentanol, 1-propanol, isopropyl alcohol, teterhydrofuran,toluene, water, and 1-octanol. The experimental solubility data wasrepresented well with the NRTL-SAC model.

FIG. 16 includes a graph illustrating the experimental solubilities vs.calculated solubilities for estriol in 9 solvents at 298.15K. Theexperimental solubility data was represented well with the NRTL-SACmodel. The data for tetrahydrofuran is found to be a very significantoutlier and it is not included in the 9 solvents shown in FIG. 16.

FIG. 17 includes a graph illustrating the experimental solubilities vs.calculated solubilities for estrone in various solvents at 298.15K. Theexperimental solubility data was represented well with the NRTL-SACmodel.

FIG. 18 includes a graph illustrating the experimental solubilities vs.calculated solubilities for morphine in 6 solvents at 308.15K.Cyclohexane and hexane were outliers. They are very low solubilitysolvents for morphine and the quality of the data is possibly subject tolarger uncertainties.

FIG. 19 illustrates a graph of the experimental solubilities vs.calculated solubilities for piroxicam in 14 solvents at 298.15K.1,2-dichloroethane, chloroform, diethyl ether, and N,N-dimethylformamide(DMF) were found to be major outliers and are not included in the 14solvents shown in FIG. 19. Interestingly, Bustamante, et al. alsoreported 1,2-dichloroethane, chloroform, and diethyl ether as outliersin their study based on solubility parameter models. P. Bustamante, etal., Partial Solubility Parameters of Piroxicam and Niflumic Acid, 1998INT. J. OF PHARM. 174, 141.

FIG. 20 illustrates a graph of the experimental solubilities vs.calculated solubilities for hydrocortisone in 11 solvents at 298.15K.Hexane is excluded because of the extreme low solubility ofhydrocortisone in hexane which could possibly subject the data to largeruncertainty.

FIG. 21 illustrates a graph of the experimental solubilities vs.calculated solubilities for haloperidol in 13 solvents at 298.15K.Haloperidol showed unusually high solubilities in chloroform and DMF andthese two solvents are not included in the 13 solvents.

The average RMS error on ln x for the predictions vs. experimentalsolubility data in Table 4 is 0.37. This corresponds to about ±45%accuracy in solubility predictions.

Example 3 Comparison of NRTL-SAC Model to Prior Art Methods forPharmaceutical Components

The solubilities of various pharmaceutical compounds was modeled withthe NRTL-SAC approach of the present invention as well as some prior artmodels (e.g., the Hanson model and the UNIFAC model) to compare theirrelative accuracies. The pharmaceutical compounds used included VIOXX®,ARCOXIA®, Lovastatin, Simvastatin, FOSAMAX®. (Available from Merck &Co., Inc., Whitehouse Station, N.J.). The solvents used included water,N,N-Dimethylformamide (“DMF”), 1-propanol, 2-propanol, 1-butanol,toluene, Chloro-benzene, acetonitrile, ethyl acetate, methanol, ethanol,heptane, acetone, and triethylamine (TEA).

Saturated solutions of the compounds in the solvents were allowed toequilibrate for at least 48 hours. Supernatant fluid was filtered anddiluted, and a high pressure liquid chromatography (HPLC) concentrationanalysis was performed to compare the predicted solubility values withactual solubility values. The NRTL-SAC model of the present inventiongave a RMS error on ln x of about 0.5 (i.e., an accuracy and predictivecapability of ±˜50%), while the Hansen model had a RMS error on ln x ofmore than 0.75 and the UNIFAC model had a RMS error on ln x of more than1.75. Additional comparisons were made for dual-solvent/pharmaceuticalsystems, and acceptable predictions were obtained from the NRTL-SACmodel of the present invention.

These experiments show that the NRTL-SAC model is a simple correlativeactivity coefficient equation that requires only component-specificmolecular parameters (i.e., numbers and types of conceptual segments).Conceptually, the approach suggests that a practitioner account for theliquid ideality of both small solvent molecules and complexpharmaceutical molecules in terms of component-specific molecularparameters (e.g., hydrophobicity, polarity, and hydrophilicity). Inpractice, these molecular parameters become the adjustable parametersthat are determined from selected experimental data. With thedevelopment of molecular parameters for solvents and organic solutes,engineering calculations can be performed for various phase equilibriumstudies, including solubilities in solvents and solvent mixtures forsolvent selection. The NRTL-SAC model provides good qualitativerepresentation on phase behaviors of organic solvents and their complexpharmaceutical solutes and it offers a practical predictive methodologyfor use in pharmaceutical process design.

Example 4 NRTL Segment Activity Coefficient Model

The NRTL-SAC activity coefficient model for component I is composed ofthe combinatorial term γ_(I) ^(C) and the residual term γ_(I) ^(R):

lnγ_(I)=lnγ_(I) ^(C)+lnγ_(I) ^(R)  (20)

Here the combinatorial term γ_(I) ^(C) is calculated from theFlory-Huggins equation for the combinatorial entropy of mixing. Theresidual term γ_(I) ^(R) is calculated from the local composition (lc)interaction contribution γ_(I) ^(lc) of Polymer NRTL (Chen, C.-C., “ASegment-Based Local Composition Model for the Gibbs Energy of PolymerSolutions,” Fluid Phase Equilibria, 83:301, 1993) (herein “Chen 1993”).The Polymer NRTL equation incorporates the segment interaction conceptand computes activity coefficient for component I in a solution bysumming up contributions to activity coefficient from all segments thatmake up component I. The equation is given as follows:

$\begin{matrix}{\mspace{79mu} {{{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m}^{{lc},I}}} \right\rbrack}}}}\mspace{20mu} {with}}} & (21) \\{{\ln \; \Gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{j}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}} & (22) \\{{\ln \; \Gamma_{m}^{{lc},I}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j,I}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} \right)}}}} & (23) \\{\mspace{76mu} {x_{j} = {{\frac{\sum\limits_{I}{x_{I}r_{j,I}}}{\sum\limits_{I}{\sum\limits_{i}{x_{I}r_{i,I}}}}\mspace{14mu} x_{j,I}} = \frac{r_{j,I}}{\sum\limits_{j}r_{j,I}}}}} & (24)\end{matrix}$

where I is the component index, i,j,k,m,m′ are the segment speciesindex, x_(I) is the mole fraction of component I, x is the segment-basedmole fraction of segment species j, r_(m,I) is the number of segmentspecies m contained only in component I, Γ_(m) ^(lc) is the activitycoefficient of segment species m, and Γ_(m) ^(lc,I) is the activitycoefficient of segment species m contained only in component I. G and τin Eqs. 22 and 23 are local binary quantities related to each other bythe NRTL non-random factor parameter α:

G=exp(−ατ).  (25)

Four pre-defined conceptual segments were suggested by Chen and Song(2004 above and in parent patent application): one hydrophobic (x), twopolar (y− and y+), and one hydrophilic (z). The model molecularparameters, i.e., hydrophobicity X, polarity types Y− and Y+, andhydrophilicity Z, correspond to r_(m,I) (m=x, y−, y+, z), numbers ofvarious conceptual segments in component I.

In the notation used throughout this disclosure, subscript I (uppercase) refers to components while subscript i (lower case) refers tosegments.

Example 5 eNRTL Segment Activity Coefficient Model

The extension of NRTL-SAC model for electrolytes is based on thegeneralized eNRTL model as summarized by Chen and Song (Chen, C.-C. andY. Song, “Generalized Electrolyte NRTL Model for Mixed-SolventElectrolyte Systems,” AIChE J., 50:1928, 2004b; herein incorporated byreference) (herein “Chen, 2004b). Here Applicants briefly present thegeneralized eNRTL model followed by details of the extended NRTL-SACmodel of the present invention.

The generalized eNRTL model is applied to correlate mean ionic activitycoefficient of mixed-solvent electrolyte systems. The segmentinteraction concept provides the framework to explicitly account for theattractive interaction of ions with the hydrophilic segments of organicsolvents and the repulsive interaction of ions with the hydrophobicsegments of organic solvents. In the generalized eNRTL model, anycomponent, electrolyte or solvent, can be defined as an oligomerconsisting of various segment species. For instance, an organicelectrolyte species can be defined as an oligomer consisting of cationicsegment, anionic segment and molecular segment. An organic solvent canbe also defined as an oligomer consisting of multiple molecular segmentsof different nature. Accordingly, with the conventional activitycoefficient accounting for the local interaction (Chen, 1993) and thelong-range interaction, the model that uses the unsymmetricPitzer-Debye-Hückel (PDH) formula (Pitzer, K. S., “Electrolytes: FromDilute Solutions to Fused Salts,” J. Am. Chem. Soc., 102, 2902 (1980))(herein “Pitzer, 1980”) is calculated as follows:

$\begin{matrix}{\begin{matrix}{{\ln \; \gamma_{I}^{*}} = {\frac{1}{RT}\left( \frac{\partial G_{m}^{*{ex}}}{\partial n_{I}} \right)_{T,P,n_{i \neq j}}}} \\{= {{\frac{1}{RT}\left( \frac{\partial G_{m}^{{*{ex}},{lc}}}{\partial n_{I}} \right)_{T,P,n_{i \neq j}}} + {\frac{1}{RT}\left( \frac{\partial G_{m}^{{*{ex}},{PDH}}}{\partial n_{I}} \right)_{T,P,n_{i \neq j}}}}}\end{matrix}{or}} & (26) \\{{\ln \; \gamma_{I}^{*}} = {{\ln \; \gamma_{I}^{*{lc}}} + {\ln \; \gamma_{I}^{*{PDH}}}}} & (27)\end{matrix}$

where I is the component index, “*” denotes the unsymmetric convention,γ_(I) is the activity coefficient of the component I in the mixture; Ris the gas constant; T is the temperature; P is the pressure; and n_(I)is the mole number of the component I in the mixture. The unsymmetricPDH formula, G*_(m) ^(ex,PDH), is obtained by normalization to molefractions of unity for solvents and zero for electrolytes (Pitzer, K.S., “Thermodynamics of Electrolytes. I: Theoretical and GeneralEquations,” J. Phys. Chem., 77, 268 (1973)). The local interaction NRTLmodel, G_(m) ^(ex,lc), is developed as a symmetric model (Chen, C.-C.,“A Segment-Based Local Composition Model for the Gibbs Energy of PolymerSolutions,” Fluid Phase Equilib., 83, 301 (1993); and Chen, C.-C., C. P.Bokis, and P. M. Mathias, “A Segment-Based Excess Gibbs Energy Model forAqueous Organic Electrolyte Systems,” AIChE J., 47, 2593 (2001)), basedon the symmetrical reference state so that the derived activitycoefficient, is γ_(I) ^(lc)=1 as x_(I)→1 for any component (species).The model is then normalized by the unsymmetric reference state (thatis, the infinite-dilution activity coefficient in an aqueous ormixed-solvent solution) to obtain the unsymmetric model, G*_(m)^(ex,lc). Accordingly, the unsymmetric convention activity coefficientis calculated as follows:

$\begin{matrix}{{{\ln \; \gamma_{I}^{*{lc}}} = {{\ln \; \gamma_{I}^{lc}} - {\ln \; \gamma_{I}^{\infty \; {lc}}}}}{{{\ln \; \gamma_{I}^{*{lc}}} = {{{\ln \; \gamma_{I}^{lc}} - {\ln \; \gamma_{I}^{\infty \; {lc}}\ln \; \gamma_{I}^{lc}}} = {\frac{1}{RT}\left( \frac{\partial G_{m}^{{ex},{lc}}}{\partial n_{I}} \right)_{T,P,n_{i \neq j}}}}},\gamma_{I}^{\infty}}} & (28) \\{{{\ln \; \gamma_{I}^{lc}} = {\frac{1}{RT}\left( \frac{\partial G_{m}^{{ex},{lc}}}{\partial n_{I}} \right)_{T,P,n_{i \neq j}}}},} & (29)\end{matrix}$

where γ_(I) ^(∞) is the infinite-dilution activity coefficient of theionic component I in an aqueous or mixed-solvent solution as calculatedby Equation 23. A more detailed description on the generalizedelectrolyte-NRTL model is depicted in Chen, 2004b.

This generalized segment interaction concept is advantageous when onemust exactly account for the different interaction characteristics thatmay be attributed to different molecules, solvents or solutes. Theability to exactly account for such different segment-segmentinteractions between different species in a system is shown to be keyfor quantitative correlation of mean ionic activity coefficients inmixed-solvent electrolyte systems. In the generalized eNRTL model,however, it is necessary to account for an electrolyte segment for eachand every species separately. Therefore, in a system that involvesmultiple components, there could be tens of different segments toconsider and hundreds of segment-segment interactions to account for,and the computation for activity coefficients becomes much morecomplicated.

Derived from and improved upon the generalized eNRTL model, theelectrolyte extension of NRTL model of the present invention providesone conceptual electrolyte segment. A “conceptual electrolyte segment”herein is one predefined electrolyte segment that characterizes theprominent interaction mechanisms between molecules in the liquid phase,that account for the liquid phase nonideality. This pre-definedelectrolyte segment is used as a reference against which all electrolytesegments are measured in terms of their liquid phase interactioncharacteristics. Unlike the generalized eNRTL model, which has no such“conceptual electrolyte segment” as a reference point, surfaceinteraction characteristics of electrolyte segments of the presentinvention are normalized against the “conceptual electrolyte segment”(in a preferred embodiment, one with interaction characteristics ofNaCl) and mathematically expressed as an equivalent number of thereference one. Having a point of reference for the calculation of theelectrolyte segment provides a unified and consistent description ofliquid phase nonideality of all electrolyte segments and a moreintuitive and powerful predictive tool in modeling physical propertiesincluding solubility. Together with the numbers of “conceptual”hydrophobic segment, hydrophilic segment and polar segment, the numberof “conceptual electrolyte segment” reflects the nature of the surfaceinteractions and their characteristic surface interaction areas thatdetermine their phase behavior.

In the simplest case of a strong electrolyte CA, one may use thefollowing chemical reaction to describe the complete dissociation of theelectrolyte:

CA→υ _(C) C ^(Z) ^(C) +υ_(A) A ^(Z) ^(A)   (30) with

υ_(C) Z _(C)=υ_(A) Z _(A)  (31)

where υ_(C) is the cationic stoichiometric coefficient, υ_(A) is theanionic stoichiometric coefficient, Z_(C) is the absolute charge numberfor cation C, and Z_(A) is the absolute charge number for anion A.

In applying the segment contribution concept to electrolytes, Applicantsintroduce a new conceptual electrolyte segment e. This conceptualsegment e would completely dissociate to a cationic segment (c) and ananionic segment (a), both of unity charge. Applicants then follow thelike-ion repulsion and the electroneutrality constraints imposed by thegeneralized eNRTL model to derive the activity coefficient equations forionic segments c and a. All electrolytes, organic or inorganic,symmetric or unsymmetric, univalent or multivalent, are to berepresented with this conceptual uni-univalent electrolyte segment etogether with previously defined hydrophobic segment, x, polar segments,y− and y+, and hydrophilic segment, z. Due to the fact that Applicantsintroduce only one (a universally useable one) conceptual electrolytesegment e, the resulting eNRTL-SAC model of the present invention ismuch simpler than the generalized eNRTL model proposed earlier.

Example 6 Solubility of an Electrolyte

Described below is the solubility of an electrolyte by the expression:

$\begin{matrix}{{{K_{sp}(T)} = {\prod\limits_{C}\; {x_{C}^{v_{C},{SAT}}\gamma_{C}^{*_{v_{C},{SAT}}}{\prod\limits_{A}\; {x_{A}^{v_{A},{SAT}}\gamma_{A}^{*_{v_{A},{SAT}}}{\prod\limits_{M}\; {x_{M}^{SAT}\gamma_{M}^{SAT}}}}}}}},} & (32)\end{matrix}$

where Ksp is the solubility product constant for the electrolyte, T isthe temperature of the mixture, x_(C) ^(ν) ^(C) ^(SAT) is the molefraction of a cation derived from the electrolyte at saturation point ofthe electrolyte, x_(A) ^(ν) ^(A) ^(SAT) is the mole fraction of an anionderived from the electrolyte at saturation point of the electrolyte,x_(M) ^(ν) ^(M) ^(SAT) is the mole fraction of a neutral moleculederived from the electrolyte at saturation point of the electrolyte,γ*_(C) ^(ν) ^(C) ^(,SAT) is the activity coefficient of a cation derivedfrom the electrolyte at the saturation concentration, γ*_(A) ^(ν) ^(A)^(,SAT) is the activity coefficient of an anion derived from theelectrolyte at the saturation concentration γ*_(M) ^(ν) ^(M) ^(,SAT) isthe activity coefficient of a neutral molecule derived from theelectrolyte at the saturation concentration, C is the cation, A is theanion, M is solvent or solute molecule, T is the temperature of themixture, y* is the unsymmetric activity coefficient of a species insolution, SAT is saturation concentration, υ_(C) is the cationicstoichiometric coefficient, υ_(A) is the anionic stoichiometriccoefficient, and υ_(M) is the neutral molecule stoichiometriccoefficient.

A major consideration in the extension of NRTL-SAC for electrolytes isthe treatment of reference state for activity coefficient calculations.While the conventional reference state for nonelectrolyte systems is thepure liquid component, the conventional reference state for electrolytesin solution is the infinite-dilution aqueous solution and thecorresponding activity coefficient is “unsymmetric.”

Following the generalized eNRTL model, the logarithm of unsymmetricactivity coefficient of an ionic species, lnγ*_(I), is the sum of threeterms: the local composition term, lnγ*_(I) ^(lc), thePitzer-Debye-Hückel term, lnγ*_(I) ^(PDH), and the Flory-Huggins term,lnγ*_(I) ^(FH).

lnγ*_(I)=lnγ*_(I) ^(lc)+lnγ*_(I) ^(PDH)+lnγ*_(I) ^(FH)  (33)

Eq. 33 applies to aqueous electrolyte systems where water is a solesolvent within the solution. For mixed-solvent solutions, the Born term,Δ lnγ*_(I) ^(Born), is used to correct the change of the infinitedilution reference state from the mixed-solvent composition to theaqueous solution for the Pitzer-Debye-Hückel term:

lnγ*_(I)=lnγ*_(I) ^(lc)+lnγ*_(I) ^(PDH)+lnγ*_(I) ^(FH)+lnγ*_(I)^(Born)  (34)

Since Applicants adopt the aqueous phase infinite dilution referencestate for lnγ*_(I), the Born term correction is required for non-aqueoussystems.

With the introduction of the conceptual electrolyte segment e and thecorresponding conceptual ionic segments c and a, one can rewrite Eq. 34in terms of contributions from all conceptual segments:

$\begin{matrix}\begin{matrix}{{\ln \; \gamma_{I}^{*}} = {{\ln \; \gamma_{I}^{*{lc}}} + {\ln \; \gamma_{I}^{*{PDH}}} + {\ln \; \gamma_{I}^{*{FH}}} + {\Delta \; \ln \; \gamma_{I}^{Born}}}} \\{= {{\sum\limits_{m}{r_{m,I}\left( {{\ln \; \Gamma_{m}^{*{lc}}} + {\ln \; \Gamma_{m}^{*{PDH}}}} \right)}} +}} \\{{{r_{c,I}\left( {{\ln \; \Gamma_{c}^{*{lc}}} + {\ln \; \Gamma_{c}^{*{PDH}}} + {\Delta \; \ln \; \Gamma_{c}^{Born}}} \right)} +}} \\{{{r_{a,I}\left( {{\ln \; \Gamma_{a}^{*{lc}}} + {\ln \; \Gamma_{a}^{*{PDH}}} + {\Delta \; \ln \; \Gamma_{a}^{Born}}} \right)} + {\ln \; \gamma_{I}^{*{FH}}}}}\end{matrix} & (35)\end{matrix}$

where r is the segment number, m is the conceptual molecular segmentindex (i.e., m=x, y−, y+, z), c and a are cationic and anionic segments,respectively, resulting from the dissociation of the conceptualelectrolyte segment e. Also notice that in Eq. 35, unlike the localcomposition term and the long range ion-ion interaction terms, theFlory-Huggins term remains as the component-based contribution.

For systems of single electrolyte CA with a segment number r_(e), r_(c)and r_(a) must satisfy electroneutrality and they can be computed fromr_(e), Z_(C), and Z_(A).

r _(c,C) =r _(e,CA) Z _(C)  (36)

r _(a,A) =r _(e,CA) Z _(A)  (37)

For systems of multiple electrolytes, the mixing rule is needed tocompute segment number r_(c) and r_(a) for each cation C and anion A.

$\begin{matrix}{r_{c,C} = {\sum\limits_{A}{r_{e,{CA}}{Z_{C}\left( {x_{A}{Z_{A}/{\sum\limits_{A^{\prime}}{x_{A^{\prime}}Z_{A^{\prime}}}}}} \right)}}}} & (38) \\{r_{a,A} = {\sum\limits_{C}{r_{e,{CA}}{Z_{A}\left( {x_{C}{Z_{C}/{\sum\limits_{C^{\prime}}{x_{C^{\prime}}Z_{C^{\prime}}}}}} \right)}}}} & (39)\end{matrix}$

r_(e,CA), the number of conceptual electrolyte segment e in electrolyteCA, becomes the new model parameter for electrolytes. For the sake ofbrevity, Applicants call r_(e,CA) parameter E, the electrolyte segmentnumber.

Example 7 Local Composition Interaction Contribution

To derive the expression for the local composition interactioncontribution, Applicants simplify the generalized excess Gibbs energyexpression of the prior Chen and Song model (Chen, 2004b) for systemswith multiple molecular segments m and single electrolyte segment e. Thesingle electrolyte segment e is then decomposed into a cationic segmentc and an anionic segment a:

$\begin{matrix}{\frac{G^{{ex},{lc}}}{nRT} = {\sum\limits_{I}\begin{bmatrix}{{\sum\limits_{m}{r_{m,I}{x_{I}\left( \frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} \right)}}} +} \\{{r_{c,I}{x_{I}\left( \frac{\sum\limits_{j}{x_{j}G_{{jc},{ac}}\tau_{{jc},{ac}}}}{\sum\limits_{k}{x_{k}G_{{kc},{ac}}}} \right)}} + {r_{a,I}{x_{I}\left( \frac{\sum\limits_{j}{x_{j}G_{{ja},{ca}}\tau_{{ja},{ca}}}}{\sum\limits_{k}{x_{k}G_{{ka},{ca}}}} \right)}}}\end{bmatrix}}} & (40) \\{\mspace{79mu} {with}} & \; \\{\mspace{79mu} {{x_{j} = {\frac{\sum\limits_{I}{x_{I}r_{j,I}}}{\sum\limits_{I}{\sum\limits_{i}{x_{I}r_{i,I}}}}\mspace{14mu} i}},{j = m},c,a}} & (41)\end{matrix}$

where G^(ex,lc) is the excess Gibbs energy from local compositioninteractions, n is the total mole number, R is the gas constant and T isthe temperature.

To derive the segment activity coefficient, one can rewrite Eq. 40 asfollows:

$\begin{matrix}{\frac{G^{{ex},{lc}}}{n_{S}{RT}} = {{\sum\limits_{m}{x_{m}\left( \frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} \right)}} + {x_{c}\left( \frac{\sum\limits_{j}{x_{j}G_{{jc},{ac}}\tau_{{jc},{ac}}}}{\sum\limits_{k}{x_{k}G_{{kc},{ac}}}} \right)} + {x_{a}\left( \frac{\sum\limits_{j}{x_{j}G_{{ja},{ca}}\tau_{{ja},{ca}}}}{\sum\limits_{k}{x_{k}G_{{ka},{ca}}}} \right)}}} & (42)\end{matrix}$

where n_(S) is the total number of all segments. Accordingly, thesegment activity coefficient can be calculated as follows:

$\begin{matrix}{{{\ln \; \Gamma_{j}^{lc}} = {\frac{1}{RT}\left( \frac{\partial G^{{ex},{lc}}}{\partial n_{j}} \right)_{T,P,n_{i \neq j}}}}{i,{j = m},c,a}} & (43)\end{matrix}$

Specifically, the activity coefficients from Eq. 43 for molecularsegments, cationic segment, and anionic segment can be carried out asfollows:

$\begin{matrix}{{\ln \; \Gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{m\; m^{\prime}}}{x_{k}G_{k\; m^{\prime}}}\left( {\tau_{m\; m^{\prime}} - \frac{\sum\limits_{j}^{\;}{x_{j}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m^{\prime}}}}} \right)}} + {\frac{x_{c}G_{{m\; c},{a\; c}}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m}}}\left( {\tau_{{m\; c},{a\; c}} - \frac{\sum\limits_{j}^{\;}{x_{j}G_{{jc},{a\; c}}\tau_{{jc},\; {a\; c}}}}{\sum\limits_{k}^{\;}{x_{k}G_{{kc},{a\; c}}}}} \right)} + {\frac{x_{a}G_{{ma},{ca}}}{\sum\limits_{k}^{\;}{x_{k}G_{{ka},{ca}}}}\left( {\tau_{{ma},{ca}} - \frac{\sum\limits_{j}^{\;}{x_{j}G_{{ja},{ca}}\tau_{{ja},{ca}}}}{\sum\limits_{k}^{\;}{x_{k}G_{{ka},{ca}}}}} \right)}}} & (44) \\{{\ln \; \Gamma_{c}^{lc}} = {{\sum\limits_{m}^{\;}{\frac{x_{m}G_{c\; m}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m}}}\left( {\tau_{c\; m} - \frac{\sum\limits_{j}^{\;}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m}}}} \right)}} + \frac{\sum\limits_{j}^{\;}{x_{j}G_{{jc},{a\; c}}\tau_{{jc},{a\; c}}}}{\sum\limits_{k}^{\;}{x_{k}G_{{kc},{a\; c}}}} - {\frac{x_{a}}{\sum\limits_{k}^{\;}{x_{k}G_{{ka},{ca}}}}\left( \frac{\sum\limits_{j}^{\;}{x_{j}G_{{ja},{ca}}\tau_{{ja},{ca}}}}{\sum\limits_{k}^{\;}{x_{k}G_{{ka},{ca}}}} \right)}}} & (45) \\{{\ln \; \Gamma_{a}^{lc}} = {{\sum\limits_{m}^{\;}{\frac{x_{m}G_{am}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m}}}\left( {\tau_{am} - \frac{\sum\limits_{j}^{\;}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}^{\;}{x_{k}G_{k\; m}}}} \right)}} + \frac{\sum\limits_{j}^{\;}{x_{j}G_{{ja},{ca}}\tau_{{ja},{ca}}}}{\sum\limits_{k}^{\;}{x_{k}G_{{ka},{ca}}}} - {\frac{x_{c}}{\sum\limits_{k}^{\;}{x_{k}G_{{kc},{a\; c}}}}\left( \frac{\sum\limits_{m}^{\;}{x_{m}G_{{m\; c},{a\; c}}\tau_{{m\; c},{a\; c}}}}{\sum\limits_{k}^{\;}{x_{k}G_{{kc},{a\; c}}}} \right)}}} & (46)\end{matrix}$

The local composition term for the logarithm of activity coefficient ofcomponent I is computed as the sum of the individual segmentcontributions.

$\begin{matrix}\begin{matrix}{{{\ln \; \gamma_{I}^{lc}} = {{\sum\limits_{i}^{\;}{r_{i,I}\ln \; \Gamma_{i}^{lc}\mspace{14mu} i}} = m}},c,a} \\{= {{\sum\limits_{m}^{\;}{r_{m,I}\ln \; \Gamma_{m}^{lc}}} + {r_{c,I}\ln \; \Gamma_{c}^{lc}} + {r_{a,I}\ln \; \Gamma_{a}^{lc}}}}\end{matrix} & (47)\end{matrix}$

However, the activity coefficient by Eq. 47 needs to be furthernormalized so that γ*_(I) ^(lc)=1 as x_(I)→1 for any component; this isthe so-called symmetric reference state. The normalization can be doneas follows:

$\begin{matrix}\begin{matrix}{{{\ln \; \gamma_{I}^{lc}} = {{\sum\limits_{i}^{\;}{{r_{i,I}\left\lbrack {{\ln \; \Gamma_{i}^{lc}} - \ln_{i}^{{lc},I}} \right\rbrack}\mspace{14mu} i}} = m}},c,a} \\{= {{\sum\limits_{m}^{\;}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m}^{{lc},I}}} \right\rbrack}} + r_{c,l}}} \\{{\left\lbrack {{\ln \; \Gamma_{c}^{lc}} - {\ln \; \Gamma_{c}^{{lc},I}}} \right\rbrack + {r_{a,I}\left\lbrack {{\ln \; \Gamma_{a}^{lc}} - {\ln \; \Gamma_{a}^{{lc},I}}} \right\rbrack}}}\end{matrix} & (48)\end{matrix}$

Here Γ*_(i) ^(lc,I) is the activity coefficient of the segment icontained in the symmetric reference state of component I; it can becalculated from Eqs. 44-46 by setting x_(I)=1:

lnΓ*_(i) ^(lc,I)=lnΓ*_(I) ^(FH)(x _(I)=1)i=m,c,a  (49)

Finally, the unsymmetric convention in Eq. 34 requires us to compute theinfinite-dilution activity coefficient, γ*_(I) ^(∞lc), for a component:

$\begin{matrix}{{\ln \; \gamma_{I}^{*{lc}}} = {{\ln \; \gamma_{I}^{lc}} - {\ln \; \gamma_{I}^{\infty \; {lc}}\mspace{14mu} {with}}}} & (50) \\\begin{matrix}{{{\ln \; \gamma_{I}^{\infty \; {lc}}} = {{\sum\limits_{i}^{\;}{{r_{i,I}\left\lbrack {{\ln \; \Gamma_{i}^{\infty \; {lc}}} - {\ln \; \Gamma_{i}^{{lc},I}}} \right\rbrack}\mspace{14mu} i}} = m}},c,a} \\{= {{\sum\limits_{m}^{\;}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{\infty \; {lc}}} - {\ln \; \Gamma_{m}^{{lc},I}}} \right\rbrack}} + r_{c,I}}} \\{{\left\lbrack {{\ln \; \Gamma_{c}^{\infty \; {lc}}} - {\ln \; \Gamma_{c}^{{lc},I}}} \right\rbrack + {r_{a,I}\left\lbrack {{\ln \; \Gamma_{a}^{\infty \; {lc}}} - {\ln \; \Gamma_{a}^{{lc},I}}} \right\rbrack}}}\end{matrix} & (51)\end{matrix}$

Combining Eqs. 48 and 51, one can obtain:

$\begin{matrix}{\begin{matrix}{{\ln \; \gamma_{I}^{*{lc}}} = {{\ln \; \gamma_{I}^{lc}} - {\ln \; \gamma_{I}^{\infty \; {lc}}}}} \\{{= {{\sum\limits_{i}^{\;}{{r_{i,I}\left\lbrack {{\ln \; \Gamma_{i}^{lc}} - {\ln \; \Gamma_{i}^{\infty \; {lc}}}} \right\rbrack}\mspace{14mu} i}} = m}},c,a} \\{= {{\sum\limits_{m}^{\;}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m}^{\infty \; {lc}}}} \right\rbrack}} + {r_{c,I}\left\lbrack {{\ln \; \Gamma_{c}^{lc}} - {\ln \; \Gamma_{c}^{\infty \; {lc}}}} \right\rbrack} +}} \\{{r_{a,I}\left\lbrack {{\ln \; \Gamma_{a}^{lc}} - {\ln \; \Gamma_{a}^{\infty \; {lc}}}} \right\rbrack}} \\{= {{\sum\limits_{m}^{\;}{r_{m,I}\ln \; \Gamma_{m}^{*{lc}}}} + {r_{c,I}\ln \; \Gamma_{c}^{*{lc}}} + {r_{a,I}\ln \; \Gamma_{a}^{*{lc}}}}}\end{matrix}{with}} & (52) \\{{\ln \; \Gamma_{m}^{*{lc}}} = {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m}^{\infty \; {lc}}}}} & (53) \\{{\ln \; \Gamma_{c}^{*{lc}}} = {{\ln \; \Gamma_{c}^{lc}} - {\ln \; \Gamma_{c}^{\infty \; {lc}}}}} & (54) \\{{\ln \; \Gamma_{a}^{*{lc}}} = {{\ln \; \Gamma_{a}^{lc}} - {\ln \; \Gamma_{a}^{\infty \; {lc}}}}} & (55)\end{matrix}$

Because Applicants adopt the aqueous phase infinite dilution referencestate, the infinite-dilution activity coefficients of conceptualsegments can be calculated from Eqs. 44-47 by setting x_(w)=1:

lnγ*_(i) ^(∞lc)=lnγ*_(i) ^(lc)(x _(w)=1)i=m,c,a  (56)

where x_(w) is the mole fraction of water in the solution.

Example 8 Long-Range Interaction Contribution from Pitzer-Debye-Hückel(PDH) Model

To account for the long-range ion-ion interactions, the presentinvention eNRTL-SAC model uses the unsymmetric Pitzer-Debye-Hückel (PDH)formula (Pitzer, 1980) on the segment basis:

$\begin{matrix}{{\frac{G^{{*{ex}},{PDH}}}{n_{S}{RT}} = {{- \left( \frac{1000}{{\overset{\_}{M}}_{S}} \right)^{1/2}}\left( \frac{4A_{\phi}I_{x}}{\rho} \right){\ln \left( {1 + {\rho \; I_{x}^{1/2}}} \right)}}}{with}} & (57) \\{A_{\phi} = {\frac{1}{3}\left( \frac{2\pi \; N_{A}{\overset{\_}{d}}_{S}}{1000} \right)^{1/2}\left( \frac{Q_{e}^{2}}{{\overset{\_}{ɛ}}_{S}k_{B}T} \right)^{3/2}}} & (58) \\{I_{x} = {\frac{1}{2}{\sum\limits_{i}^{\;}{x_{i}z_{i}^{2}}}}} & (59)\end{matrix}$

where A_(φ) is the Debye-Hückel parameter, I_(x) is the ionic strength(segment mole fraction scale), M _(S) is the average molecular weight ofthe mixed-solvents, ρ is the closest approach parameter, N_(A) is theAvogadro's number, d _(S) is the average density of the mixed-solvents,Q_(∈) is the electron charge, ∈ _(S) is the average dielectric constantof the mixed-solvents, k_(B) is the Boltzmann constant, and z_(i)(z_(m)=0; z_(c)=z_(a)=1) is the charge number of segment-based speciesi.

Applying the PDH model to the conceptual segments, the activitycoefficient of segment species i can be derived as follows:

$\begin{matrix}\begin{matrix}{{{\ln \; \Gamma_{i}^{*{PDH}}} = {\frac{1}{RT}\left( \frac{\partial G^{{*{ex}},{PDH}}}{\partial n_{i}} \right)_{T,P,n_{j \neq i}}\mspace{14mu} i}},{j = m},c,a} \\{= {{- \left( \frac{1000}{{\overset{\_}{M}}_{S}} \right)^{1/2}}{A_{\phi}\left\lbrack {{\left( \frac{2z_{i}^{2}}{\rho} \right){\ln \left( {1 + {\rho \; I_{x}^{1/2}}} \right)}} + \frac{{z_{i}^{2}I_{x}^{1/2}} - {2I_{x}^{3/2}}}{1 + {\rho \; I_{x}^{1/2}}}} \right\rbrack}}}\end{matrix} & (60)\end{matrix}$

The unsymmetric long range term for the logarithm of activitycoefficient of component I is the sum of contributions from its varioussegments:

$\begin{matrix}{\mspace{79mu} {{{\ln \; \gamma_{I}^{*{PDH}}} = {{\sum\limits_{m}^{\;}{r_{m,I}\ln \; \Gamma_{m}^{*{PDH}}}} + {r_{c,I}\ln \; \Gamma_{c}^{*{PDH}}} + {r_{a,I}\ln \; \Gamma_{a}^{*{PDH}}}}}\mspace{85mu} {where}}} & (61) \\{\mspace{79mu} {{\ln \; \Gamma_{m}^{*{PDH}}} = {2\left( \frac{1000}{{\overset{\_}{M}}_{S}} \right)^{1/2}\frac{A_{\phi}I_{x}^{3/2}}{1 + {\rho \; I_{x}^{1/2}}}}}} & (62) \\{{{{\ln \; \Gamma_{c}^{*{PDH}}} = {{\ln \; \Gamma_{a}^{*{PDH}}} = {{- \left( \frac{1000}{{\overset{\_}{M}}_{S}} \right)^{1/2}}{A_{\phi}\left\lbrack {{\left( \frac{2}{\rho} \right){\ln \left( {1 + {\rho \; I_{x}^{1/2}}} \right)}} + \frac{I_{x}^{1/2} - {2I_{x}^{1/2}}}{1 + {\rho \; I_{x}^{1/2}}}} \right\rbrack}}}}\; {With}}\mspace{76mu}} & (63) \\{\mspace{85mu} {A_{\phi} = {\frac{1}{3}\left( \frac{2\pi \; N_{A}{\overset{\_}{d}}_{S}}{1000} \right)^{1/2}\left( \frac{Q_{e}^{2}}{{\overset{\_}{ɛ}}_{S}k_{B}T} \right)^{3/2}}}} & (64) \\{\mspace{79mu} {I_{x} = {\frac{1}{2}\left( {x_{c} + x_{a}} \right)}}} & (65)\end{matrix}$

The Debye-Hückel theory is based on the infinite dilution referencestate for ionic species in the actual solvent media. For systems withwater as the only solvent, the reference state is the infinite dilutionaqueous solution. For mixed-solvent systems, the reference state forwhich the Pitzer-Debye-Hückel formula remains valid is the infinitedilution solution with the corresponding mixed-solvent composition.Consequently, the molecular quantities for the single solvent need to beextended for mixed-solvents; simple composition average mixing rules areadequate to calculate them as follows:

$\begin{matrix}{{\overset{\_}{M}}_{S} = {\sum\limits_{S}^{\;}{x_{S}^{\prime}M_{S}}}} & (66) \\{\frac{1}{{\overset{\_}{d}}_{S}}{\sum\limits_{S}^{\;}\frac{x_{S}^{\prime}}{d_{S}}}} & (67) \\{{{\overset{\_}{ɛ}}_{S} = {\sum\limits_{S}^{\;}{w_{S}^{\prime}ɛ_{S}}}}{with}} & (68) \\{x_{S}^{\prime} = \frac{x_{S}}{\sum\limits_{S}^{\;}x_{S}}} & (69) \\{w_{S}^{\prime} = \frac{M_{S}x_{S}}{\sum\limits_{S}^{\;}{M_{S}x_{S}}}} & (70)\end{matrix}$

where S is a solvent component in the mixture, and M_(s) is themolecular weight of the solvent S. It should be pointed out that Eqs.66-70 should be used only in Eq. 54 and M _(S), d _(S), and ∈ _(S) werealready assumed as constants in Eqs. 57 and 58 when deriving Eq. 60 formixed-solvent systems. Table 1 shows the values of dielectric constantat 298.15 K used in this study for the same sixty-two solventsinvestigated by Chen and Song (Chen, 2004a and U.S. Publication No.2005/0187748) above. These values were compiled from various sourcesincluding internet websites and commercial software Aspen Propertiesv2004.1 (by Aspen Technology, Inc. of Cambridge, Mass., assignee of thepresent invention).

Example 9 Born Term Correction to Activity Coefficient

Given that the infinite dilution aqueous solution is chosen as thereference state, one needs to correct the change of the reference statefrom the mixed-solvent composition to aqueous solution for thePitzer-Debye-Hückel term. The Born term (Robinson, R. A. and R. H.Stokes, Electrolyte Solutions, 2^(nd) ed., Butterworths (1970), Rashin,A. A. and B. Honig, “Reevaluation of the Born Model of Ion Hydration, J.Phys. Chem., 89: 5588 (1985)) on the segment basis is used for thispurpose:

$\begin{matrix}{\frac{\Delta \; G^{Born}}{n_{S}{RT}} = {\frac{Q_{e}^{2}}{2k_{B}T}\left( {\frac{1}{{\overset{\_}{ɛ}}_{S}} - \frac{1}{ɛ_{W}}} \right){\sum\limits_{i}^{\;}{\frac{x_{i}z_{i}^{2}}{r_{i}}10^{- 2}}}}} & (71)\end{matrix}$

ΔG^(Born) is the Born term correction to the unsymmetricPitzer-Debye-Hückel formula, G*^(ex,PDH), ∈_(W) is the dielectricconstant of water, and r, is the Born radius of segment specie i.

Applying Eq. 65 to all conceptual segments, the corresponding expressionfor the activity coefficient of segment species i can be derived asfollows:

$\begin{matrix}{{{\Delta \; \ln \; \Gamma_{m}^{Born}} = {{\frac{1}{RT}\left( \frac{{\partial\Delta}\; G^{Born}}{\partial n_{m}} \right)_{T,P,n_{j \neq m}}} = {{0\mspace{14mu} m} = x}}},{y -},{y +},z} & (72) \\{{{{\Delta ln}\; \Gamma_{i}^{Born}} = {{\frac{1}{RT}\left( \frac{{\partial\Delta}\; G^{Born}}{\partial n_{i}} \right)_{T,P,n_{j \neq i}}} = {\frac{Q_{e}^{2}}{2k_{B}T}\left( {\frac{1}{{\overset{\_}{ɛ}}_{S}} - \frac{1}{ɛ_{W}}} \right)\frac{z_{i}^{2}}{r_{i}}10^{- 2}}}}\mspace{25mu} \mspace{76mu} {{i = c},a}} & (73)\end{matrix}$

The Born correction term on the logarithm of activity coefficient ofcomponent I is the sum of contributions from its various segments:

$\begin{matrix}{{\Delta \; \ln \; \gamma_{I}^{Born}} = {{r_{c,I}\Delta \; \ln \; \Gamma_{c}^{Born}} + {r_{a,I}\Delta \; \ln \; \Gamma_{a}^{Born}}}} & (74) \\{{\Delta \; \ln \; \gamma_{c}^{Born}} = {\frac{Q_{e}^{2}}{2k_{B}T}\left( {\frac{1}{{\overset{\_}{ɛ}}_{S}} - \frac{1}{ɛ_{W}}} \right)\frac{1}{r_{c}}10^{- 2}}} & (75) \\{{\Delta \; \ln \; \gamma_{a}^{Born}} = {\frac{Q_{e}^{2}}{2k_{B}T}\left( {\frac{1}{{\overset{\_}{ɛ}}_{S}} - \frac{1}{ɛ_{W}}} \right)\frac{1}{r_{a}}10^{- 2}}} & (76)\end{matrix}$

Example 10 Flory-Huggins Term Correction to Activity Coefficient

Although in most common electrolyte systems, the combinatorial entropyof mixing term is much smaller than the residual term, one may stillwant to include it in a general model. Applicants follow the PolymerNRTL model (Chen 1993 above) and use the Flory-Huggins term to describethe combinatorial term:

$\begin{matrix}{{\frac{G^{{ex},{FH}}}{nRT} = {\sum\limits_{I}^{\;}{x_{I}{\ln \left( \frac{\varphi_{I}}{x_{I}} \right)}}}}{with}} & (77) \\{\varphi_{I} = \frac{x_{I}r_{I}}{\sum\limits_{J}^{\;}{x_{J}r_{J}}}} & (78)\end{matrix}$

where G^(ex,FH) is the Flory-Huggins term for the excess Gibbs energy,φ_(I), is the segment fraction of component I, and r_(I) is the numberof all conceptual segments in component I:

$\begin{matrix}{r_{I} = {{\sum\limits_{m}^{\;}r_{m,I}} + r_{c,I} + r_{a,I}}} & (79)\end{matrix}$

The activity coefficient of component I from the combinatorial term canbe derived from Eq. 66:

$\begin{matrix}{{\ln \; \gamma_{I}^{FH}} = {{{\ln \left( \frac{\varphi_{I}}{x_{I}} \right)} + 1 - {r_{I}{\sum\limits_{J}^{\;}\frac{\varphi_{J}}{r_{J}}}}} = {{\ln\left( \frac{r_{I}}{\sum\limits_{J}^{\;}{x_{J}r_{J}}} \right)} + 1 - \frac{r_{I}}{\sum\limits_{J}^{\;}{x_{J}r_{J}}}}}} & (80)\end{matrix}$

The infinite-dilution activity coefficient of a component in water is:

$\begin{matrix}{{\ln \; \gamma_{I}^{\infty \; {FH}}} = {{\ln \left( \frac{r_{I}}{r_{W}} \right)} + 1 - \frac{r_{I}}{r_{W}}}} & (81)\end{matrix}$

In both NRTL-SAC (parent patent application) and present inventioneNRTL-SAC, water is selected as the reference for the hydrophilicsegment z. Therefore, one can set r_(w)=1. Thus, one has:

lnγ_(I) ^(∞FH)=ln r _(I)+1−r _(I)  (82)

One can then compute the unsymmetric activity coefficient from theFlory-Huggins term as follows:

$\begin{matrix}{{\ln \; \gamma_{I}^{*{,{FH}}}} = {{{\ln \; \gamma_{I}^{FH}} - {\ln \; \gamma_{I}^{\infty \; {FH}}}} = {r_{I} - {\ln\left( {\sum\limits_{J}{x_{J}r_{J}}} \right)} - \frac{r_{I}}{\sum\limits_{J}{x_{J}r_{J}}}}}} & (83)\end{matrix}$

Example 11 NRTL Binary Parameters

In Eqs. 20 and 21 for NRTL-SAC, the model formulation requires theasymmetric interaction energy parameters, τ, and the symmetric nonrandomfactor parameters, α, for each binary pair of the conceptual segments.In Eqs. 44-46 for eNRTL-SAC of the present invention, one needsadditional binary parameters of τ and α between conceptual molecularsegments, m and ionic segments, c or a. In practice, Applicants fix thevalues of α's for the binary pairs of molecular segment and ionicsegment to the single value of 0.2 while the values of τ for the binarypairs of molecular segment and ionic segment are calculated from the τ'sfor the binary pairs of molecular segment and electrolyte segment.Following the same scheme in generalized eNRTL (Chen and Song, 2004babove), one can calculate these binary interaction energy parameters asfollows:

τ_(cm)=τ_(am)=τ_(em)  (84)

τ_(mc,ac)=τ_(ma,ca)=τ_(me)  (85)

Following the treatment of NRTL-SAC (disclosed in U.S. Publication No.2005/0187748), Applicants identify a reference electrolyte for theconceptual electrolyte segment e. In searching for the referenceelectrolyte, Applicants choose one elemental electrolyte that hasabundant literature data. In one example study, NaCl is used as thereference electrolyte for e. The ionic radii for sodium ion and chlorideion are 1.680×10⁻¹⁰ m and 1.937×10⁻¹⁰ m, respectively. With NaCl as thereference electrolyte, the energy parameters for the z-e pair are set to(8.885, −4.549) for the water-NaCl pair. The energy parameters for thex-e pair are set to (15, 5), in line with the parameters identified forC₂H₄—NaCl pair earlier by Chen and Song (Chen, 2004b). The energyparameters for the y-e pairs are set to (12, −3) after limited trials tooptimize the performance of the model in this study. The complete set ofNRTL binary interaction energy parameters are given in Table 6. Otherchoices of the reference electrolyte and parameter values may besuitable. The below reports the general behavior of the presentinvention eNRTL-SAC model based on the parameters reported in Table 6.

The electrolyte segment e is the only extra molecular descriptor and theelectrolyte parameter E is the only extra molecular parameter for allelectrolytes, inorganic or organic. All local and long rangeinteractions derived from the existence of cationic and anionic speciesof various ionic charge valence, radius, chemical make-up, etc., are tobe accounted for with this extra molecular descriptor for electrolytestogether with combinations of conceptual molecular segments, i.e.,hydrophobicity, polarity and hydrophilicity. In other words, everyelectrolyte, organic or inorganic, are modeled as combinations of E, X,Y, and Z. As such, electrolytes are recognized as “hydrophobic”electrolytes, “polar” electrolytes, “hydrophilic” electrolytes, andtheir various combinations. Likewise, ionic activity coefficient of eachionic species will be computed from its share of E, X, Y, and Z. Theions are to be considered as “hydrophobic” ions, “polar” ions, or“hydrophilic” ions.

FIGS. 22 to 26 show effects of the molecular parameters on mean ionicactivity coefficients (mole fraction scale) of the referenceelectrolyte, i.e., electrolyte with E=1. As shown in FIGS. 22 to 26,hydrophobicity parameter X brings down the mean ionic activitycoefficient at low electrolyte concentration but in a rather nonlinearway. Polarity parameter Y− raises the mean ionic activity coefficientwhile polarity parameter Y+ lowers the mean ionic activity coefficient.Hydrophilicity parameter Z has a relatively slight downshift effect onthe mean ionic activity coefficient. Electrolyte parameter E brings downthe mean ionic activity coefficient at low electrolyte concentration andpushes up the mean ionic activity coefficient at high electrolyteconcentration.

Experimental data for ionic activity coefficients are not readilyavailable though emerging (Wilczek-Vera, G. et al, “On the Activity ofIons and the Junction Potential: Revised Values for All Data,” AIChE J.,50:445, 2004). Given the fact that existing experimental data arelimited to mean ionic activity coefficient for neutral electrolytes,Applicants are not able to directly identify the molecular parametersfor ionic species. In preparing FIGS. 22 to 26 discussed above and thesubsequent studies reported in the Model Applications section below,Applicants use Eqs. 36-37 to determine from electrolyte parameter E theionic segment numbers for the ions and Applicants arbitrarily assignmolecular segment parameters (X, Y−, Y+, and Z) only to the anion. Thispractice is acceptable since virtually all electrolytes investigated inthis study are electrolytes with elemental cations.

Limited amount of mean ionic activity coefficient data are available inthe public literature for aqueous electrolytes. Applicants test theeNRTL-SAC model 20 as shown in FIG. 3 against mean ionic activitycoefficient data of aqueous electrolyte systems. In addition, Applicantstest the eNRTL-SAC model against salt solubility data in multiplesolvents for a number of inorganic electrolytes and organicelectrolytes. To the best of Applicants' knowledge, public literaturedata is very scarce for such salt solubility data. Proprietarysolubility data from industrial collaborators was also used to test theapplicability of the eNRTL-SAC model. However, results with suchproprietary solubility data are not included in this discussion.

Example 12 Mean Ionic Activity Coefficients in Aqueous Systems

For an electrolyte CA that dissociates to cation C and anion A, the meanionic activity coefficients γ*_(±) is related to individual ionicactivity coefficients as follows:

$\begin{matrix}{{\ln \; \gamma_{\pm}^{*}} = {\frac{1}{v}\left( {{v_{C}\ln \; \gamma_{C}^{*}} + {v_{A}\ln \; \gamma_{A}^{*}}} \right)}} & (86)\end{matrix}$

where ν=ν_(C)+ν_(A).

Equation 83 gives the mean ionic activity coefficient on the molefraction scale and it can be converted to the molality scale:

lnγ*_(±m)=lnγ*_(±)−ln(1+vmM_(S)/1000)  (87)

where γ*_(±m) is the mean ionic activity coefficient on the molalityscale, m is the molality of the salt (mol/kg-solvent), and M_(S) is themolecular weight of the solvent (g/mol).

Table 7 shows the fit to molality scale mean ionic activity coefficientdata and the identified electrolyte and molecular parameters for theaqueous inorganic and organic electrolytes at 298.15 K as compiled by ofRobinson and Stokes (1970) cited above. All mean ionic activitycoefficient data are assumed to have standard deviation of 5%. The datafor C5 and higher sodium carboxylates were excluded from the fit becausethese organic electrolytes were known to form micelles at highelectrolyte concentrations (Chen, C.-C. et al., “Segment-Based ExcessGibbs Energy Model for Aqueous Organic Electrolytes, AIChE J., 47:2593,2001). With a few exceptions such as LiBr, most uni-univalent anduni-bivalent electrolytes are well represented as combinations of E andY− or Y+ parameters. Most uni-univalent electrolytes have E parameteraround unity while higher E values are found for higher valentelectrolytes. Applicants also found that the fit seems to deterioratefor electrolytes with higher E values, This observation is consistentwith the understanding that higher valent electrolytes are known toprone to the formation of hydrated species or other complexationspecies. The relatively poor representation of these electrolytes withthe model reflects the inadequate assumption of complete dissociationfor such electrolytes (Chen, C.-C.; et al., “Unification of Hydrationand Dissociation Chemistries with the Electrolyte NRTL Model,” AIChEJournal, 45:1576, 1999). As a derived property, mean ionic activitycoefficient becomes meaningless if the complete dissociation assumptionof electrolytes does not hold true.

To illustrate the quality of the fit, FIG. 27 shows the comparison ofexperimental and calculated molality scale mean ionic activitycoefficients for five aqueous electrolytes at 298.15 K. The solid linesare the calculated values from the model. It shows that the eNRTL-SACmodel provides reasonable qualitative representation of the data whilethe original eNRTL model (Chen, C.-C. et al., “Local Composition Modelfor Excess Gibbs Energy of Electrolyte Systems,” AIChE J., 28:588, 1982)achieves excellent quantitative representation of the data.

Example 13 Salt Solubility in Mixed Solvent Systems

At the solubility limit of a nonelectrolytes, the solubility productconstant, K_(sp), can be written in terms of the product of the soluteconcentration and the solute activity coefficient at the saturationconcentration:

K _(sp) =x _(I)γ_(I)  (88)

At the solubility limit of an electrolyte, ionic species precipitate toform salt.

ν_(C) C ^(Z) ^(C+ν) _(A) A ^(Z) ^(A) →C _(ν) _(C) A _(ν) _(A)_((S))  (89)

The corresponding solubility product constant can be defined as follows.

K _(sp) =x _(C) ^(ν) ^(C) γ*_(C) ^(ν) ^(C) x _(A) ^(ν) ^(A) γ*_(A) ^(ν)^(A)   (90)

Eqs. 89 and 90 can be expanded to include solvent molecules and otherspecies if the solid polymorph involves hydrates, othersolvent-containing salts, double salts, triple salts, and others.

Applicants tested the applicability of eNRTL-SAC with the very limitedpublic literature data and some proprietary data on solubilities of anumber of inorganic and organic electrolytes in various solvents. Thisdescription presents the results with solubility data from publicliterature. To bring certain consistency to the data treatment,Applicants convert all solute solubility data to mole fraction (exceptfor sodium chloride and sodium acetate). Applicants also assign standarddeviation of 10% to all solute solubility data within range of 1 to 0.1,standard deviation of 20% to all solute solubility data with range of0.1 to 0.01, standard deviation of 30% to data with range of 0.01 to0.001, and so on.

Solubility data of sodium chloride in twelve different solvents at298.15 K were successfully fitted with the eNRTL-SAC model. (Note thatthe temperature for the acetone data is 291.15 K and the temperature forthe ethyl acetate data is 292.15 K. However, they are included as ifthey were data at 298.15 K.) The sodium chloride solubilities in thetwelve solvents vary by six orders of magnitude. The satisfactory fit ofthe data for ten solvents (formic acid and ethyl acetate excluded) isshown in FIG. 28. The eNRTL-SAC model predicts one order-of-magnitudehigher solubility for sodium chloride in formic acid and virtually nosolubility for sodium chloride in ethyl acetate while the data suggestsvery low but measurable solubility. The molecular parameters and thesolubility product constant were adjusted simultaneously to provide thebest fit to the data and the identified values are given in Table 8. InTable 8, the last column to the right quantifies goodness of fit todata. It is worth noting that the electrolyte parameter E for sodiumchloride is near unity, similar to the parameters reported in Table 3for sodium chloride.

Solubility data of sodium acetate in five different solvents was alsofitted successfully with the eNRTL-SAC model. The solubilities in thefive solvents vary by four orders of magnitude. The fit of the data isshown in FIG. 29. The solid phase for the solubility measurements isanhydrous sodium acetate. Note that the data for methanol and acetonewas taken at 291.15 K while the data for water and ethylene glycol wastaken at 298.15 K. The temperature for the 1-propanol data is not known.In fitting the data, Applicants treated all data as if it was 298.15 Kdata. The identified molecular parameters and the solubility productconstant are given in Table 8. As an organic electrolyte, theelectrolyte parameter E for sodium acetate is found to be significantlyless than unity.

FIGS. 30 a and 30 b show satisfactory representations of the solubilitydata of benzoic acid in twenty-six solvents (Beerbower, A. et al.,“Expanded Solubility Parameter Approach. I. Naphthalene and Benzoic Acidin Individual Solvents,” J. Pharm. Sci., 73:179, 1984) and thesolubility data of sodium benzoate in ten solvents (Bustamante, P. etal., “The Modified Extended Hansen Method to Determine PartialSolubility Parameters of Drugs Containing a Single Hydrogen BondingGroup and Their Sodium Derivatives: Benzoic Acid/Na and Ibuprofen/Na,”Int. J. of Pharmaceutics, 194:117, 2000). These solvents are chosen inthis study because of the availability of the NRTL-SAC parameters forthe solvents from Applicants' prior work. The identified molecularparameters for the two solutes were given in Table 8. It is interestingthat the molecular parameters identified for benzoic acid withtwenty-six solvents in this study are quite similar to the molecularparameters identified for benzoic acid with seven solvents inApplicants' earlier study. Applicants also noted that the solubilityrange expands as benzoic acid is converted to sodium benzoate.Furthermore, the molecular parameters have changed from ahydrophobic/polar/hydrophilic combination (benzoic acid) to apolar/hydrophilic/electrolytic combination (sodium benzoate). Solubilitydata of sodium benzoate in seven other solvents (chloroform, benzene,dioxane, cyclohexane, ethyl acetate, heptane and chlorobenzene) isexcluded from FIG. 29 b because the eNRTL-SAC model predicts virtuallyno solubility for sodium benzoate in these solvents while the datasuggests very low but measurable solubility. It is probable that themolecular form of sodium benzoate may be present in such highlyhydrophobic solvents. However, due to their low concentrations,Applicants chose to ignore these low solubility solvents in this studyalthough the current thermodynamic framework can be used to account forthe two solubility routes, i.e., Eqs. 88 and 90, individually orsimultaneously.

FIGS. 31 a and 31 b show successful representations of the solubilitydata of salicylic acid in eighteen solvents and the solubility data ofsodium salicylate in thirteen solvents (Barra, J. et al., “Propositionof Group Molar Constants for Sodium to Calculate the Partial SolubilityParameters of Sodium Salts Using the van Krevelen Group ContributionMethod,” Eur. J. of Pharm. Sci., 10:153, 2000). Their molecularparameters were given in Table 8. Like the molecular parameters forbenzoic acid and the sodium salt, the molecular parameters have changedfrom a hydrophobic/polar/hydrophilic combination (salicylic acid) to apolar/hydrophilic/electrolytic combination (sodium salicylate).Solubility data of sodium salicylate in benzene, cyclohexane, andheptane is excluded from FIG. 31 b, again because the eNRTL-SAC modelpredicts virtually no solubility of sodium salicylate in these threesolvents although the data suggests very low but measurable solubility.Acetic acid is the only outlier among solvents with significantsolubility for sodium salicylate. The eNRTL-SAC model prediction for thesolubility of sodium salicylate in acetic acid is about one order ofmagnitude too high. Acetic acid is not included in the thirteen solventsshown in FIG. 31 b.

The eNRTL-SAC model results for the solubility data of p-aminobenzoicacid in nineteen solvents and sodium p-aminobenzoate in twelve solvents(Barra et al., 2000, above) are given in FIGS. 32 a and 32 b. Again, lowsolubility solvents (benzene, cyclohexane and heptane) are excluded fromFIG. 32 b for sodium aminobenzoate. Acetone and DMF are two outliers forsodium aminobenzoate and they are also excluded from FIG. 32 b. TheeNRTL-SAC model predicts two orders of magnitude higher solubilities inthese two solvents.

The solubility data and model calculations for ibuprofen in nineteensolvents and sodium ibuprofen in eleven solvents (Bustamante et al.,2000 above) are given in FIGS. 33 a and 33 b. In comparison to otherorganic solutes, one embodiment of the eNRTL-SAC model provides a ratherpoor fit to the ibuprofen data albeit a better fit than in priornonelectrolyte models. Applicants did notice that the ibuprofensolubility data from Bustamante et al. are significantly different fromthose reported by Gracin and Rasmuson (Gracin, S. and A. C. Rasmuson,“Solubility of Phenylacetic Acid, p-Hydroxyphenylacetic Acid,p-Aminophenylacetic acid, p-Hydroxybenzoic acid, and Ibuprofen in PureSolvents,” J. Chem. Eng. Data, 47:1379, 2002) for certain commonsolvents including methanol, ethanol, acetone and ethyl acetate. Noattempt was made to reconcile the differences between the Bustamantedata and the Gracin and Rasmuson data. The eNRTL-SAC model fit to thesodium ibuprofen solubility data appears to be more satisfactory. Again,the eleven solvents reported in FIG. 33 b do not include low solubilitysolvents (benzene, cyclohexane, heptane, and chlorobenzene). Similarly,acetone and DMF are two outliers for sodium ibuprofen and they are alsoexcluded from FIG. 33 b. The eNRTL-SAC model predicts two orders ofmagnitude higher solubilities in these two solvents than the availabledata. Bustamante et al. (2000, above) reported high water content of theibuprofen sample (3.3 wt % water) and the sodium ibuprofen sample (13 wt% water). It is not clear how such high water contents in the samplescould impact on the solubility measurements.

The solubility data for diclofenac in sixteen solvents and sodiumdiclofenac in ten solvents (Barra et al., 2000 above) are fitted andreported in FIGS. 34 a and 34 b. The eNRTL-SAC model significantlyoverestimates the solubilities of diclofenac in acetic acid, formamideand ethylene glycol. These three solvents are excluded from the sixteensolvents shown in FIG. 34 a. Data for low solubility solvents (benzene,cyclohexane, ethyl acetate, heptane and chlorobenzene) for sodiumdiclofenac are excluded from FIG. 34 b. Acetic acid and acetone are twooutliers with the model estimations one to three orders of magnitudehigher solubilities for sodium diclofenac. The two solvents are notincluded in FIG. 34 b.

The solubility data treatment above assumes complete dissociation ofelectrolytes and considers the solubility problem as formation of saltsfrom ionized species of electrolytes, i.e., Eq. 90. One may argue thatelectrolytes do not dissociate completely into ionic species especiallyin organic solvents of low dielectric constant. In the absence ofdissociation to ionic species, the solubility relationship can bedescribed by Eq. 88 and the eNRTL-SAC model of the present inventionreduces to the NRTL-SAC model of the parent patent application.Applicants have treated the electrolyte systems above as nonelectrolytes(i.e., no dissociation to ionic species) with NRTL-SAC and the modelresults are also included in Table 8. With the absence of electrolyteparameter, the representation of the solubility data deterioratessubstantially. Applicants also noted that the identified molecularparameters (X, Y−, Y+, and Z) with the complete dissociation treatmentare roughly twice as large as those reported with the non-dissociationtreatment. This finding is consistent with the fact that Applicants onlyassign the molecular parameters (X, Y−, Y+, and Z) to the anion,

TABLE 5 Dielectric Constant of Solvents at 298.15 K solvent namedielectric constant at 298.15 K Acetic acid 6.13 Acetone 20.83Acetonitrile 36.97 Anisole 4.3 Benzene 2.27 1-Butanol 17.7 2-Butanol15.8 n-Butyl-acetate 5.1 Methyl-tert-butyl-ether 2.6Carbon-tetrachloride 2.23 Chlorobenzene 5.56 Chloroform 4.7 Cumene 2.22Cyclohexane 2.02 1,2-Dichloroethane 10.19 1,1-Dichloroethylene 4.61,2-Dichloroethylene 4.6 Dichloromethane 8.9 1,2-Dimethoxyethane notavailable N,N-Dimethylacetamide not available N,N-Dimethylformamide 38.3Dimethyl-sulfoxide 47.2 1,4-Dioxane 2.21 Ethanol 24.11 2-Ethoxyethanolnot available Ethyl-acetate 6.02 Ethylene-glycol 41.2 Diethyl-ether 4.26Ethyl-formate 7.16 Formamide 109.5 Formic-acid 58.5 n-Heptane 1.92n-Hexane 1.89 Isobutyl-acetate 5.6 Isopropyl-acetate not availableMethanol 32.62 2-Methoxyethanol not available Methyl-acetate 6.683-Methyl-1-butanol 14.7 2-Hexanone 14.6 Methylcyclohexane 2.02Methyl-ethyl-ketone 18.5 Methyl-isobutyl-ketone 13.1 Isobutanol 17.9N-Methyl-2-pyrrolidone 33 Nitromethane 6.26 n-Pentane 1.84 1-Pentanol13.9 1-Propanol 20.1 Isopropyl-alcohol 19.9 n-propyl-acetate 6 Pyridine2.3 Sulfolane 43.3 Tetrahydrofuran 7.52 1,2,3,4-Tetrahydronaphthalenenot available Toluene 2.36 1,1,1-Trichloroethane 7.5 Trichloroethylene3.42 m-Xylene 2.24 Water 78.54 Triethylamine 2.44 1-Octanol 10.3

TABLE 6 NRTL Binary Interaction Parameters Segment (1) x x y− y+ xSegment (2) y− z z z y+ τ₁₂ 1.643 6.547 −2.000 2.000 1.643 τ₂₁ 1.83410.949 1.787 1.787 1.834 α₁₂ = α₂₁ 0.2 0.2 0.3 0.3 0.2 Segment (1) x y−y+ z Segment (2) e e e e τ₁₂ 15 12 12 8.885 τ₂₁ 5 −3 −3 −4.549 α₁₂ = α₂₁0.2 0.2 0.2 0.2

TABLE 7 Results of Fit for Molality Scale Mean Ionic ActivityCoefficient Data of Aqueous Electrolytes at 298.15 K (Data from Robinsonand Stokes, 1970) E Y− Y+ σ^(□) max. molality 1-1 Electrolytes AgNO₃0.738 1.758 0.050 6.0 CsAc 1.002 0.438 0.011 3.5 CsBr 0.950 0.678 0.0135.0 CsCl 0.948 0.643 0.014 6.0 CsI 0.956 0.719 0.012 3.0 CsNO₃ 0.9811.328 0.005 1.4 CsOH 0.942 0.354 0.002 1.0 HBr 1.135 0.654 0.034 3.0 HCl1.324 0.524 0.087 6.0 HClO₄ 1.476 0.569 0.136 6.0 HI 1.117 0.824 0.0353.0 HNO₃ 0.971 0.211 0.005 3.0 KAc 0.998 0.386 0.009 3.5 KBr 0.910 0.3110.011 5.5 KBrO₃ 0.968 1.141 0.002 0.5 KCl 0.920 0.370 0.010 4.5 KClO₃0.958 1.053 0.003 0.7 KCNS 0.876 0.477 0.019 5.0 KF 0.987 0.042 0.0044.0 KH Malonate 0.846 0.920 0.022 5.0 KH Succinate 0.912 0.665 0.011 4.5KH₂PO₄ 0.970 1.362 0.006 1.8 KI 0.903 0.168 0.011 4.5 KNO₃ 0.856 1.4610.027 3.5 KOH 1.236 0.344 0.058 6.0 K Tol 0.750 1.296 0.026 3.5 LiAc0.962 0.097 0.002 4.0 LiBr 1.422 0.526 0.116 6.0 LiCl 1.282 0.436 0.0846.0 LiClO₄ 1.145 0.681 0.047 4.0 LiI 1.058 0.712 0.033 3.0 LiNO₃ 1.0500.294 0.022 6.0 LiOH 1.028 0.652 0.022 4.0 LiTol 0.881 0.392 0.014 4.5NaAc 0.978 0.301 0.005 3.5 NaBr 0.992 0.115 0.008 4.0 NaBrO₃ 0.923 0.8020.010 2.5 Na Butyrate 0.989 0.566 0.009 3.5 NaCl 1.000 0.017 6.0 NaClO₃0.891 0.507 0.011 3.5 NaClO₄ 0.894 0.267 0.010 6.0 NaCNS 0.925 0.1280.006 4.0 NaF 0.976 0.425 0.002 1.0 Na Formate 0.905 0.094 0.013 3.5 NaHMalonate 0.878 0.664 0.019 5.0 NaH Succinate 0.924 0.495 0.010 5.0NaH₂PO₄ 0.864 1.256 0.020 6.0 NaI 1.009 0.266 0.012 3.5 NaNO₃ 0.8250.842 0.029 6.0 NaOH 1.080 0.109 0.039 6.0 Na Propionate 0.992 0.4480.006 3.0 Na Tol 0.793 0.920 0.026 4.0 NH₄Cl 0.884 0.424 0.019 6.0NH₄NO₃ 0.813 1.128 0.043 6.0 RbAc 1.012 0.416 0.011 3.5 RbBr 0.914 0.5190.016 5.0 RbCl 0.929 0.466 0.012 5.0 RbI 0.925 0.520 0.014 5.0 RbNO₃0.815 1.611 0.038 4.5 TlAc 0.864 0.952 0.033 6.0 TlClO₄ 1.020 1.2310.000 0.5 TlNO3 1.069 1.692 0.003 0.4 1-2 Electrolytes Cs₂SO₄ 1.1612.568 0.050 1.8 K₂CrO₄ 1.048 2.738 0.075 3.5 K₂SO₄ 1.386 2.475 0.021 0.7Li₂SO₄ 1.138 2.177 0.051 3.0 Na₂CrO₄ 1.091 2.443 0.051 4.0 Na₂ Fumarate1.259 1.770 0.041 2.0 Na₂ Maleate 1.202 2.699 0.075 3.0 Na₂SO₄ 0.9883.273 0.090 4.0 Na₂S₂O₃ 1.071 2.709 0.064 3.5 (NH₄)₂SO₄ 1.006 3.4770.118 4.0 Rb₂SO₄ 1.150 2.743 0.052 1.8 1-3 Electrolytes K₃Fe(CN)₆ 1.3284.996 0.101 1.4 1-4 Electrolytes K₄Fe(CN)₆ 1.449 9.448 0.146 0.9 2-1Electrolytes BaAc₂ 1.016 0.997 0.128 3.5 BaBr₂ 1.267 0.358 0.018 2.0BaCl₂ 1.227 0.585 0.029 1.8 Ba(ClO₄)₂ 1.305 0.261 0.049 5.0 BaI2 1.3540.028 0.017 2.0 Ba(NO₃)₂ 1.435 1.268 0.008 0.4 CaBr₂ 1.969 0.171 0.4956.0 CaCl₂ 1.701 0.309 0.283 6.0 Ca(ClO₄)₂ 2.021 0.431 6.0 CaI₂ 1.4190.131 0.036 2.0 Ca(NO₃)₂ 1.108 0.875 0.053 6.0 CdBr₂ 1.324 3.164 0.2944.0 CdCl₂ 1.052 3.047 0.315 6.0 CdI₂ 1.780 3.820 0.337 2.5 Cd(NO₃)₂1.176 0.500 0.037 2.5 CoBr₂ 1.779 0.218 5.0 CoCl₂ 1.397 0.194 0.046 4.0CoI₂ 2.260 0.488 6.0 Co(NO₃)₂ 1.444 0.296 0.113 5.0 CuCl₂ 1.033 0.4251.217 0.069 6.0 Cu(NO₃)₂ 1.409 0.416 0.117 6.0 FeCl₂ 1.319 0.255 0.0112.0 MgAc₂ 1.192 0.946 0.059 4.0 MgBr₂ 1.941 0.347 5.0 MgCl₂ 1.745 0.1440.275 5.0 Mg(ClO₄)₂ 1.988 0.162 0.303 4.0 MgI₂ 2.237 0.470 5.0 Mg(NO₃)₂1.493 0.198 0.140 5.0 MnCl₂ 1.273 0.343 0.020 6.0 NiCl₂ 1.533 0.1890.123 5.0 Pb(ClO₄)₂ 1.549 0.236 0.184 6.0 Pb(NO₃)₂ 1.129 1.964 0.083 2.0SrBr₂ 1.330 0.183 0.023 2.0 SrCl₂ 1.401 0.357 0.082 4.0 Sr(ClO₄)₂ 1.7420.034 0.261 6.0 SrI₂ 1.384 0.076 0.030 2.0 Sr(NO₃)₂ 0.978 1.250 0.0914.0 UO₂Cl₂ 1.277 0.024 0.017 3.0 UO₂(ClO₄)₂ 2.854 0.883 5.5 UO₂(NO₃)₂1.392 0.372 0.490 0.036 5.5 ZnBr₂ 0.906 0.337 0.088 6.0 ZnCl₂ 0.9530.971 0.065 6.0 Zn(ClO₄)₂ 2.045 0.130 0.318 4.0 ZnI₂ 0.868 0.132 0.1166.0 Zn(NO₃)₂ 1.518 0.214 0.176 6.0 2-2 Electrolytes BeSO₄ 1.376 4.0770.233 4.0 MgSO₄ 1.380 4.206 0.238 3.0 MnSO₄ 1.287 4.460 0.271 4.0 NiSO₄1.398 4.381 0.220 2.5 CuSO₄ 1.587 4.114 0.154 1.4 ZnSO₄ 1.339 4.4170.242 3.5 CdSO₄ 1.295 4.547 0.271 3.5 UO₂SO₄ 1.215 4.528 0.309 6.0 3-1Electrolytes AlCl₃ 1.730 0.579 0.087 1.8 CeCl₃ 1.562 0.883 0.047 1.8CrCl₃ 1.589 0.641 0.022 1.2 Cr(NO₃)₃ 1.551 0.761 0.036 1.4 EuCl₃ 1.5860.820 0.049 2.0 LaCl₃ 1.553 0.877 0.042 2.0 NdCl₃ 1.575 0.882 0.045 2.0PrCl₃ 1.562 0.892 0.042 2.0 ScCl₃ 1.636 0.709 0.041 1.8 SmCl₃ 1.5810.843 0.046 2.0 YCl₃ 1.629 0.807 0.057 2.0 3-2 Electrolytes Al₂(SO₄)₃1.354 4.886 0.222 1.0 Cr₂(SO₄)₃ 1.257 4.549 0.218 1.2 4-1 ElectrolytesTh(NO₃)₄ 1.273 1.251 0.056 5.0^(1.) σ is defined to be

$\left\lbrack {\sum\limits_{i}^{N}{\left( \frac{\gamma_{\pm i}^{*\exp} - \gamma_{\pm \; i}^{*{cal}}}{\gamma_{\pm i}^{*\exp}} \right)^{2}/N}} \right\rbrack^{1/2}$

where γ*_(±), is the mean ionic activity coefficient of electrolyte andN is the number of data used in correlations

TABLE 8 eNRTL-SAC Model Parameters for Solutes no. of solute solvents XY− Y+ Z E ln K_(sp) σ⁴ benzoic acid¹ 26 0.494 0.336 0.468  −1.714 0.292salicylic acid¹ 18 0.726 0.176 0.749  −1.624 0.774 p-aminobenzoic acid¹19 0.552 0.423 0.594 0.881  −3.348 1.206 Ibuprofen¹ 19 1.038 0.051 0.0280.318  −1.423 1.055 Diclofenac¹ 16 0.158 1.678 0.451  −3.560 0.991sodium chloride² 10 1.444 0.994  −6.252 0.783 sodium acetate²  5 1.4170.521  −6.355 0.241 sodium benzoate² 10 0.750 1.685 2.201 0.539  −7.3120.493 sodium salicylate² 13 0.845 2.417 0.090  −4.889 0.771 sodiump-aminobenzoate² 12 2.299 2.387 0.192  −8.293 1.258 sodium ibuprofen² 111.819 1.743 2.362 0.150 −17.844 0.886 sodium diclofenac² 10 0.409 3.5583.486 0.161 −14.202 0.858 sodium chloride³ 10 1.060 2.200  −3.540 0.923sodium acetate³  5 0.249 0.679  −2.277 0.281 sodium benzoate³ 10 0.1791.825  −2.978 0.699 sodium salicylate³ 13 0.373 1.572  −2.153 1.058sodium p-aminobenzoate³ 12 0.125 0.649 1.895  −3.247 1.904 sodiumibuprofen³ 11 0.270 0.394 0.823  −2.364 1.685 sodium diclofenac³ 100.454 0.124 2.493  −4.405 1.473 ¹nonelectrolytes ²electrolytes ³treatedas nonelectrolytes${\;^{4}\sigma \mspace{14mu} {is}\mspace{14mu} {defined}\mspace{14mu} {to}\mspace{14mu} {be}\mspace{14mu} \left( {\sum\limits_{i}^{N}\; {\left( {{\ln \mspace{14mu} x_{i}^{\exp}} - {\ln \mspace{14mu} x_{i}^{cal}}} \right)^{2}\text{/}N}} \right)^{1\text{/}2}\mspace{14mu} {where}\mspace{14mu} x\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {solubility}\mspace{14mu} {of}\mspace{14mu} {solute}},{i.e.},{{mole}\mspace{14mu} {fraction}\mspace{14mu} \left( {{note}\mspace{14mu} {that}\mspace{14mu} {mass}\mspace{14mu} {fraction}\mspace{14mu} {for}\mspace{14mu} {sodium}\mspace{14mu} {chloride}\mspace{14mu} {and}\mspace{14mu} {sodium}\mspace{14mu} {acetate}} \right)},{{and}\mspace{14mu} N\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {number}\mspace{14mu} {of}\mspace{14mu} {data}\mspace{14mu} {used}\mspace{14mu} {in}\mspace{14mu} {{correlations}.}}$

Example 14 Activity Coefficient Model in Chromatography

Chromatographers then can use activity coefficient models as describedherein to determine activity coefficients of one or more chemicalspecies in the mobile phase and the stationary phase and to selectsolvent types, solvent composition and elution gradients for the mobilephase and the column packing for the stationary phase as they optimizeand control selectivity and productivity of a chromatography reaction.

The methods disclosed herein, among other features, compute activitycoefficients from the combinatorial term and the residual term asdescribed below

lnγ_(I)=lnγ_(I) ^(C)+lnγ_(I) ^(R)  (102)

where γ_(I) is an activity coefficient for the one or more chemicalspecies, the mobile phase component or the stationary phase component,γ_(I) ^(C) is a combinatorial contribution to the activity coefficientfor the one or more chemical species, the mobile phase component or thestationary phase component, and γ_(I) ^(R) is a residual contribution tothe activity coefficient of the one or more chemical species, the mobilephase component or the stationary component.

$\begin{matrix}{\mspace{79mu} {{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m,I}^{lc}}} \right\rbrack}}}}} & (103) \\{{\ln \; \Gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}} & (104) \\{{\ln \; \Gamma_{m,I}^{lc}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j,I}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} \right)}}}} & (105)\end{matrix}$

where I is a component index for the one or more chemical species, themobile phase component, or the stationary phase component, each j, k, m.and m′ is a segment species index, x, is a segment-based mole fractionof segment species j, x_(j,I) is a segment-based mole fraction ofsegment species j in the I, r_(m,I) is a number of segment species mcontained only in the I, Γ_(m) ^(lc) is an activity coefficient ofsegment species m, Γ_(m) ^(lc,I) is an activity coefficient of segmentspecies m contained in the I, and G and τ are binary quantities relatedto each other by a non-random factor parameter α as in G=exp(−ατ) α andτ are the nonrandomness factor parameter and the segment-segment binaryinteraction energy parameter, respectively.

Example 15 Prediction of Activity Coefficient Using the NRTL SAC Method

To apply the NRTL SAC method described herein for the prediction ofsolute activity coefficients in chromatography operations, it isnecessary to identify the equivalent numbers for the solutes, thesolvents, and the various stationary phase types. The methodology toidentify these equivalent segment numbers are described above. Theequivalent numbers of the stationary phase component can be determinedfrom the partition coefficient data or the retention factor data of atleast four solutes of varied surface interaction characteristics withthe specific column packing. Such partition coefficient data reflect thenature of the solute-stationary phase surface interactions that are thebasis for the determination of the equivalent numbers for the columnpacking type.

Given the equivalent numbers for the solutes, solvents, and stationaryphase component, the NRTL SAC method can then be used to determinesolute activity coefficients in the mobile phase and the stationaryphase and chromatography column selectivity and productivity asfunctions of solutes, solvent types, solvent composition, column packingtypes, and temperature. For given solutes, the NRTL SAC method is usefulin selecting solvent types, solvent composition, and stationary phasecomponent. Integrated with mathematical model for chromatographyoperations, the model becomes a core component of comprehensive and1^(st) principles based chromatography unit operation model for design,optimization, and control of this ubiquitous active pharmaceuticalingredient purification equipment in the pharmaceutical industry.

Example 16 Modeling Solute Partition Coefficients in ChromatographyOperations

There are different types of chromatographic columns, e.g.,reverse-phase chromatography, normal phase chromatography, hydrophobicinteraction chromatography, hydrophilic interaction chromatography,ion-pair chromatography, ion-exchange chromatography, etc.

The magnitude of retention of a chemical specie (e.g., solute) inchromatography is measured under isocratic condition (e.g., constantmobile phase solvent composition) by the retention factor k_(I):

$\begin{matrix}{k_{I} = {\frac{C_{s}V_{s}}{C_{m}V_{m}} = \frac{t_{rI} - t_{0}}{t_{0}}}} & (91)\end{matrix}$

k, is a capacity ratio of the one or more chemical species, C_(s) is theconcentration in the stationary phase, V, is the volume of thestationary phase, C_(m) is the concentration in the mobile phase, V_(m)is the volume of the mobile phase, t_(rI) is a retention time of the oneor more chemical species, and t₀ is an elution time for an inert tracer.The retention time is related to the concentration of the one or morechemical species in the stationary phase. Once the retention time forthe one or more chemical species on the stationary phase (e.g., packing)of the stationary phase are known, the concentration of the one or morechemical species can be calculated and the NRTL-SAC model parameters(e.g., the equivalent numbers, etc.) described herein can be determinedby fitting the concentration data as the NRTL-SAL model parameters canbe determined for a solute by fitting solubility values.

For the separation of two chemical species I and j, selectivity isdefined by the ratio (α) of the capacity ratios of the solutes:

$\begin{matrix}{\alpha = \frac{k_{J}}{k_{I}}} & (92)\end{matrix}$

The retention factor k_(I) is related to the equilibrium constant,K_(I), or the Gibbs free energy change associated with solute transferfrom the mobile phase to the stationary phase as

$\begin{matrix}{k_{I} = {\frac{x_{s}V_{s}}{x_{m}V_{m}} = {K_{I}\Phi}}} & (93) \\{{\Delta \; G_{I}} = {{- {RT}}\; \ln \; K_{I}}} & (94)\end{matrix}$

K_(I) is a partition coefficient, Φ is a phase ratio, V_(s) is a volumeof the stationary phase component, V_(m) is a volume of the mobile phasecomponent, x_(s) is a concentration of the one or more chemical speciesinteracting with the stationary phase component, and x_(m) is aconcentration of the one or more chemical species interacting with themobile phase component.

$\begin{matrix}{\Phi = \frac{V_{s}}{V_{m}}} & (95) \\{K_{I} = \frac{x_{s}}{x_{m}}} & (96)\end{matrix}$

At thermodynamic equilibrium, the stationary phase solute activityshould be equal to the mobile phase solute activity.

a _(s) =a _(m)  (97)

x _(s)γ_(s) =x _(m)γ_(m)  (98)

The mobile phase solute activity coefficient γ_(m) is a function ofmobile phase solute and solvent composition while the stationary phasesolute activity coefficient γ_(s) is a function of solute and thestationary phase composition. The partition coefficient K_(I) can bederived from ratio of the activity coefficients.

$\begin{matrix}{K_{I} = \frac{\gamma_{m}}{\gamma_{s}}} & (99)\end{matrix}$

In chromatographic operations, solute concentrations are very low. γ_(m)^(∞) and γ_(s) ^(∞) are activity coefficients of the one or morechemical species at infinite dilution approximated from γ_(m) and γ_(s),γ_(m) is a mobile phase activity coefficient of the one or more chemicalspecies, and γ_(s) is a stationary phase activity coefficient of the oneor more chemical species.

$\begin{matrix}{K_{I} = \frac{\gamma_{m}^{\infty}}{\gamma_{s}^{\infty}}} & (100)\end{matrix}$

It is reported that at least for some chromatography systems, the soluteactivity coefficient in the stationary phase remain practically constantwhile the mobile phase composition is changed. If this observation holdstrue, then the partition coefficient K_(I) is controlled by the soluteinfinite dilution activity coefficient in the mobile phase.

Example 17 Separation of Two Bands in Chromatography

The separation of any two bands in the chromatogram can be variedsystematically by changing experimental conditions. Resolution, R, canbe expressed in terms of three parameters (k, α, and N) which aredirectly related to the experimental conditions:

$\begin{matrix}{R_{s} = {\frac{1}{4}\left( {\alpha - 1} \right)N^{1/2}\frac{k}{1 + k}}} & (101)\end{matrix}$

Here k is the average retention factor for the two bands, N is thecolumn plate number, and α is the separation factor defined in Eq. (92).As both k and α are functions of the solute activity coefficients in themobile phase and the stationary phase, they are determined by thoseconditions that affect activity coefficients the solutes in the mobilephase and the stationary phase: 1) composition of the mobile phase, 2)composition of the stationary phase, and 3) temperature. Changes in themobile phase in terms of solvent types and solvent composition, changesin the column packing, and changes in temperature affect the activitycoefficients of the solutes in the mobile phase and the stationary phasewhich then determine the selectivity and productivity of the column. Inother words, to optimize and control selectivity and productivity ofchromatography requires quantitative knowledge on how the soluteactivity coefficients are affected by the changes in the mobile phase,the stationary phase and the temperature.

While this invention has been particularly shown and described withreferences to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

1. A method of conducting chromatography for separating one or morechemical species from a mixture comprising: a) assigning moleculardescriptors based on molecular interaction characteristics to each of:(i) the one or more chemical species, (ii) a mobile phase component, and(iii) a stationary phase component; b) measuring respective molecularinteraction indices for the one or more chemical species, for the mobilephase component and for the stationary phase component; and c)determining respective activity coefficients of: (i) the one or morechemical species; (ii) the mobile phase component; and (iii) thestationary phase component, such that the determined respective activitycoefficients enable conducting chromatography for separating the one ormore chemical species from the mixture.
 2. The method of claim 1,wherein the one or more predetermined molecular descriptors are based onsurface molecular interaction characteristics.
 3. The method of claim 1,wherein measuring molecular interaction indices for the one or morechemical species, and the mobile phase component and the stationaryphase component includes determining the molecular interaction indicesbased on measures of effective surface areas expressing surfacemolecular interaction characteristics for each of the one or morechemical species, of the mobile phase component, and of the stationaryphase component.
 4. The method of claim 1, wherein assigning one or morepredetermined molecular descriptors based on molecular interactivecharacteristics includes assigning one or more predetermined conceptualsegment to each of the one or more chemical species, of the mobile phasecomponent and of the stationary phase component.
 5. The method of claim4 further including determining molecular interaction energy indicesassociated with each of the predetermined conceptual segments.
 6. Themethod of claim 4, wherein assigning one or more predeterminedconceptual segment to each of the one or more chemical species, of themobile phase component and of the stationary phase component includesidentifying respective pairwise segment-segment interaction parametersfor each of the predetermined conceptual segments.
 7. The method ofclaim 6, wherein the respective pairwise segment-segment interactionparameters are binary NRTL parameters.
 8. The method of claim 4, whereinassigning one or more predetermined conceptual segment to each of theone or more chemical species, of the mobile phase component and of thestationary phase component includes describing the molecular interactivecharacteristics in terms of a hydrophobic segment, a polar segment, asolvation segment or a hydrophilic segment.
 9. The method of claim 1,wherein measuring respective molecular interaction indices for the oneor more chemical species, and the mobile phase component and thestationary phase component includes determining an equivalent number forthe predetermined conceptual segment.
 10. The method of claim 1 furtherincluding separating the one or more chemical species from the mixture.11. The method of claim 1, wherein the one or more chemical species isan active pharmaceutical ingredient.
 12. The method of claim 1, whereinthe mobile phase component includes one or more solvents.
 13. The methodof claim 12, wherein the molecular interaction energy indices aresegment-segment binary interaction energy parameters.
 14. The method ofclaim 1, wherein the mobile phase component includes one or moresolvents.
 15. The method of claim 1, wherein the stationary phasecomponent includes one or more solvents and interacting surface areas.16. The method of claim 1, wherein determining activity coefficients ofthe one or more chemical species, the mobile phase component and thestationary phase component includes using the following formula:lnγ_(I)=lnγ_(I) ^(C)+lnγ_(I) ^(R), wherein: γ_(I) is an activitycoefficient for the one or more chemical species, for the mobile phasecomponent or for the stationary phase component; γ_(I) ^(C) is acombinatorial contribution to the activity coefficient for the one ormore chemical species, for the mobile phase component or for thestationary phase component; and γ_(I) ^(R) is a residual contribution tothe activity coefficient of the one or more chemical species, for themobile phase component or for the stationary phase component.
 17. Themethod of claim 15, further including the following formulas forcomputing respective active coefficients: $\begin{matrix}{\mspace{79mu} {{{\ln \; \gamma_{I}^{R}} = {{\ln \; \gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\left\lbrack {{\ln \; \Gamma_{m}^{lc}} - {\ln \; \Gamma_{m,I}^{lc}}} \right\rbrack}}}},}} \\{{{\ln \; \Gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} \right)}}}},\text{}\mspace{20mu} {and}} \\{{{\ln \; \Gamma_{m,I}^{lc}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\left( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{j}{x_{j,I}G_{{jm}^{\prime}}\tau_{{jm}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} \right)}}}},}\end{matrix}$ wherein: I is a component index for the one or morechemical species, the mobile phase component, or the stationary phasecomponent each j, k, m. and m′ is a segment species index; x_(j) is asegment-based mole fraction of segment species j; x_(j,I) is asegment-based mole fraction of segment species j in I; r_(m,I) is anequivalent number of segment species m contained only in I; Γ_(m) ^(lc)is an activity coefficient of segment species m; Γ_(m) ^(lc,I) is anactivity coefficient of segment species m contained in I; and G and τare binary quantities related to each other by a non-random factorparameter α as in G=exp(−ατ).
 18. The method of claim 1, whereinseparating the one or more chemical species includes calculating theretention time of the one or more chemical species.
 19. The method ofclaim 18, wherein calculating the retention time of the one or morechemical species includes computing a capacity ratio of the one or morechemical species.
 20. The method of claim 18, wherein the capacity ratiois a function of the molecular interaction: a) between the one or morechemical species and a mobile phase component; and b) between the one ormore chemical species and a stationary phase component.
 21. The methodof claim 19, wherein computing the capacity ratio includes using thefollowing formulas: ${k_{I} = \frac{t_{rI} - t_{0}}{t_{0}}},{and}$${k_{I} = {\frac{x_{s}V_{s}}{x_{m}V_{m}} = {K_{I}\Phi}}},$ wherein:k_(I) is a capacity ratio of the one or more chemical species; t_(rI) isa retention time of the one or more chemical species; t₀ is an elutiontime for an inert tracer; K_(I) is a partition coefficient; Φ is a phaseratio; V_(s) a volume of the stationary phase component; V_(m) is avolume of the mobile phase component; x_(s) is a concentration of theone or more chemical species interacting with the stationary phasecomponent; and x_(m) is a concentration of the one or more chemicalspecies interacting with the mobile phase component.
 22. The method ofclaim 21 further including computing the partition coefficient K_(I)using the following formula:${K_{I} = \frac{\gamma_{m}^{\infty}}{\gamma_{s}^{\infty}}},$ wherein:γ_(m) ^(∞) and γ_(s) ^(∞) are activity coefficients of the one or morechemical species at infinite dilution approximated from γ_(m) and γ_(s);γ_(m) is a mobile phase activity coefficient of the one or more chemicalspecies; and γ_(s) is a stationary phase activity coefficient of the oneor more chemical species.
 23. The method of claim 1, wherein usingavailable experimental data for the one or more chemical species and oneor more molecules that make up the mobile phase component and thestationary phase component includes using experimental phase equilibriumdata.
 24. The method of claim 1, wherein the one or more chemicalspecies is an active pharmaceutical ingredient.
 25. The method of claim1 further includes selecting independently a composition of the mobilephase component, a composition of the stationary phase component and atemperature for separating a desired chemical species from the mixture.26. The method of claim 1, wherein the chromatography is reverse-phasechromatography, normal phase chromatography, hydrophobic interactionchromatography, hydrophilic interaction chromatography, ion-pairchromatography, or ion-exchange chromatography.
 27. A computer programproduct, comprising: a) a computer usable medium; and b) a set ofcomputer program instructions embodied on the computer usable medium forconducting chromatography for separating a one or more chemical speciesfrom a mixture by: 1) assigning molecular descriptors based on molecularinteraction characteristics to each of: (i) the one or more chemicalspecies, (ii) a mobile phase component, and (iii) a, stationary phasecomponent; 2) measuring respective molecular interaction indices for theone or more chemical species, for the mobile phase component and for thestationary phase component; and 3) determining respective activitycoefficients of: (i) the one or more chemical species; (ii) the mobilephase component; and (iii) the stationary phase component, such that thedetermined respective activity coefficients enables conductingchromatography for separating the one or more chemical species from themixture.
 28. The computer program product of claim 27, wherein the oneor more chemical species includes an active pharmaceutical ingredient.29. The computer program product of claim 27 further including computingat least one physical property for each of the one or more chemicalspecies, the mobile phase component and the stationary phase componentusing the determined at least one conceptual segment.
 30. The computerprogram product of claim 27 further including selecting independently asolvent composition of the mobile phase component, the stationary phasecomponent and a temperature for separating a desired chemical speciesfrom the mixture.
 31. The computer program product of claim 27, whereinat least some portion of the computer program is transmitted over aglobal network.
 32. The computer program product of claim 27, whereinthe computer usable medium includes a removable storage medium.
 33. Thecomputer program product of claim 32, wherein the removable storagemedium includes any of a CD-ROM, a DVD-ROM, a diskette, and a tape. 34.A computer system for conducting chromatography comprising: a) a userinput means for obtaining empirical data from a user; b) a digitalprocessor coupled to receive the empirical data from the input means,wherein the digital processor executes a modeling system in workingmemory, wherein the modeling system uses the empirical data for: i)assigning molecular descriptors based on molecular interactioncharacteristics to each of: (i) each of the one or more chemicalspecies; (ii) a mobile phase component; and (iii) a stationary phasecomponent; ii) measuring respective molecular interactive indices forthe one or more chemical species, and the mobile phase component and forthe stationary phase component; and iii) determining activitycoefficients of: (i) the one or more chemical species; (ii) the mobilephase component; and (iii) the stationary phase component such that thedetermined respective activity coefficients enables conductingchromatography for separating the one or more chemical species from themixture.
 35. The computer system of claim 34 further comprising anoutput means coupled to the digital processor, wherein the output meansprovides to the user indications of selecting the mobile phase, and thestationary phase and, optionally, a reaction temperature for the one ormore chemical species from the modeling system.
 36. The computer systemof claim 34, wherein the computer system enables transmission of someportion of the empirical data and results from the modeling system overa global network.
 37. The computer system of claim 34, wherein thechromatography includes reverse-phase chromatography, normal phasechromatography, hydrophobic interaction chromatography, hydrophilicinteraction chromatography, ion-pair chromatography, or ion-exchangechromatography.